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Unsupervised Random Quantum Networks for PDEs (2312.14975v1)
Published 21 Dec 2023 in quant-ph and cs.LG
Abstract: Classical Physics-informed neural networks (PINNs) approximate solutions to PDEs with the help of deep neural networks trained to satisfy the differential operator and the relevant boundary conditions. We revisit this idea in the quantum computing realm, using parameterised random quantum circuits as trial solutions. We further adapt recent PINN-based techniques to our quantum setting, in particular Gaussian smoothing. Our analysis concentrates on the Poisson, the Heat and the Hamilton-Jacobi-BeLLMan equations, which are ubiquitous in most areas of science. On the theoretical side, we develop a complexity analysis of this approach, and show numerically that random quantum networks can outperform more traditional quantum networks as well as random classical networks.
- Berahas AS, Cao L, Choromanski K, et al (2022) A theoretical and empirical comparison of gradient approximations in derivative-free optimization. Foundations of Computational Mathematics 22(2):507–560 Bhuvaneswari et al [2012] Bhuvaneswari V, Lingeshwaran S, Balachandran K (2012) Weak solutions for p𝑝pitalic_p-Laplacian equation. Adv Nonlinear Anal 1:319–334 Cuomo et al [2022] Cuomo S, Di Cola VS, Giampaolo F, et al (2022) Scientific machine learning through Physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing 92(3):88 Doumèche et al [2023] Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Bhuvaneswari V, Lingeshwaran S, Balachandran K (2012) Weak solutions for p𝑝pitalic_p-Laplacian equation. Adv Nonlinear Anal 1:319–334 Cuomo et al [2022] Cuomo S, Di Cola VS, Giampaolo F, et al (2022) Scientific machine learning through Physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing 92(3):88 Doumèche et al [2023] Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Cuomo S, Di Cola VS, Giampaolo F, et al (2022) Scientific machine learning through Physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing 92(3):88 Doumèche et al [2023] Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Bhuvaneswari V, Lingeshwaran S, Balachandran K (2012) Weak solutions for p𝑝pitalic_p-Laplacian equation. Adv Nonlinear Anal 1:319–334 Cuomo et al [2022] Cuomo S, Di Cola VS, Giampaolo F, et al (2022) Scientific machine learning through Physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing 92(3):88 Doumèche et al [2023] Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Cuomo S, Di Cola VS, Giampaolo F, et al (2022) Scientific machine learning through Physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing 92(3):88 Doumèche et al [2023] Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
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In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Doumèche N, Biau G, Boyer C (2023) Convergence and error analysis of PINNs. ArXiv:2305.01240 Dunjko and Briegel [2018] Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Dunjko V, Briegel HJ (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics 81(7):074001 Evans [2022] Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. 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The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Evans LC (2022) Partial differential equations, vol 19. American Mathematical Society. Glasserman [2004] Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Glasserman P (2004) Monte Carlo Methods in Financial Engineering, vol 53. Springer Gonon [2023] Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Gonon L (2023) Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. Journal of Machine Learning Research 24(189):1–51 Gonon and Jacquier [2023] Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Gonon L, Jacquier A (2023) Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. ArXiv:2307.12904 Gonon et al [2023] Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gonon L, Grigoryeva L, Ortega JP (2023) Approximation bounds for random neural networks and reservoir systems. The Annals of Applied Probability 33(1):28–69 Gopakumar et al [2023] Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
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Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Gopakumar V, Pamela S, Samaddar D (2023) Loss landscape engineering via data regulation on PINNs. Machine Learning with Applications 12:100464 Han et al [2018] Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Han J, Jentzen A, E W (2018) Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115(34):8505–8510 He et al [2023] He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. 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Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- He D, Li S, Shi W, et al (2023) Learning Physics-informed neural networks without stacked back-propagation. In: International Conference on AI and Statistics, pp 3034–3047 Jacquier and Zuric [2023] Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Jacquier A, Zuric Z (2023) Random neural networks for rough volatility. ArXiv:2305.01035 Krishnapriyan et al [2021] Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Krishnapriyan A, Gholami A, Zhe S, et al (2021) Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems 34:26548–26560 Kyriienko et al [2021] Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Kyriienko O, Paine AE, Elfving VE (2021) Solving nonlinear differential equations with differentiable quantum circuits. Physical Review A 103:052416 Lee and Kang [1990] Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Lee H, Kang IS (1990) Neural algorithm for solving differential equations. Journal of Computational Physics 91(1):110–131 Lu et al [2021] Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Lu L, Meng X, Mao Z, et al (2021) DeepXDE: A deep learning library for solving differential equations. SIAM review 63(1):208–228 Luo et al [2020] Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Luo XZ, Liu JG, et al PZ (2020) Yao.jl: Extensible, efficient framework for quantum algorithm design. Quantum 4:341 Mari et al [2021] Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Mari A, Bromley TR, Killoran N (2021) Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A 103(1):012405 Mattheakis et al [2021] Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Mattheakis M, Joy H, Protopapas P (2021) Unsupervised reservoir computing for solving ordinary differential equations. ArXiv:2108.11417 Müller and Zeinhofer [2019] Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Müller J, Zeinhofer M (2019) Deep Ritz revisited. ArXiv:1912.03937 Pérez-Salinas et al [2020] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
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Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, et al (2020) Data re-uploading for a universal quantum classifier. Quantum 4:226 Pérez-Salinas et al [2021a] Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Pérez-Salinas A, Cruz-Martinez J, Alhajri AA, et al (2021a) Determining the proton content with a quantum computer. Physical Review D 103(3):034027 Pérez-Salinas et al [2021b] Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Pérez-Salinas A, López-Núñez D, García-Sáez A, et al (2021b) One qubit as a universal approximant. Physical Review A 104(1):012405 Preskill [2023] Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Preskill J (2023) Quantum computing 40 years later. In: Feynman Lectures on Computation. CRC Press, p 193–244 Raissi et al [2019] Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378:686–707 Wendel [1948] Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Wendel JG (1948) Note on the Gamma function. The American Mathematical Monthly 55(9):563–564 Wenshu et al [2022] Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Wenshu Z, Daolun L, Luhang S, et al (2022) Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics 54(3):543–556 Yu and E [2018] Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Yu B, E W (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1):1–12 Zoufal et al [2019] Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1) Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)
- Zoufal C, Lucchi A, Woerner S (2019) Quantum generative adversarial networks for learning and loading random distributions. npj Quantum Information 5(1)