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A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I (2307.04121v1)

Published 9 Jul 2023 in cs.LG, cond-mat.mtrl-sci, cs.NA, math.AP, and math.NA

Abstract: Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.

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References (49)
  1. Computational fluid dynamics, volume 206. Springer, 1995.
  2. A first order hyperbolic framework for large strain computational solid dynamics. part i: Total lagrangian isothermal elasticity. Computer Methods in Applied Mechanics and Engineering, 283:689–732, 2015.
  3. Jan Achenbach. Wave propagation in elastic solids. Elsevier, 2012.
  4. Rajat Arora. Computational Approximation of Mesoscale Field Dislocation Mechanics at Finite Deformation. PhD thesis, Carnegie Mellon University, 2019.
  5. Rinaldo M Colombo. Hyperbolic phase transitions in traffic flow. SIAM Journal on Applied Mathematics, 63(2):708–721, 2003.
  6. Dispersion-relation-preserving finite difference schemes for computational acoustics. Journal of computational physics, 107(2):262–281, 1993.
  7. J Blake Temple. Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics. Journal of Differential Equations, 41(1):96–161, 1981.
  8. Hyperbolic conservation laws in continuum physics, volume 3. Springer, 2005.
  9. Discontinuous Galerkin methods: theory, computation and applications, volume 11. Springer Science & Business Media, 2012.
  10. Gary A Sod. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of computational physics, 27(1):1–31, 1978.
  11. Randall J LeVeque et al. Finite volume methods for hyperbolic problems, volume 31. Cambridge university press, 2002.
  12. Upwind difference schemes for hyperbolic systems of conservation laws. Mathematics of computation, 38(158):339–374, 1982.
  13. Peter K Sweby. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM journal on numerical analysis, 21(5):995–1011, 1984.
  14. Physics-informed neural networks for modeling rate-and temperature-dependent plasticity. arXiv preprint arXiv:2201.08363, 2022.
  15. Spatio-temporal super-resolution of dynamical systems using physics-informed deep-learning. arXiv preprint arXiv:2212.04457, 2022.
  16. Physics-informed neural networks for inverse problems in supersonic flows. arXiv preprint arXiv:2202.11821, 2022.
  17. Limitations of physics informed machine learning for nonlinear two-phase transport in porous media. Journal of Machine Learning for Modeling and Computing, 1(1), 2020.
  18. Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for pdes. IMA Journal of Numerical Analysis, 42(2):981–1022, 2022.
  19. Thermodynamically consistent physics-informed neural networks for hyperbolic systems. Journal of Computational Physics, 449:110754, 2022.
  20. Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering, 360:112789, 2020.
  21. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. In AAAI Spring Symposium: MLPS, 2021.
  22. Distributed physics informed neural network for data-efficient solution to partial differential equations. arXiv preprint arXiv:1907.08967, 2019.
  23. Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 365:113028, 2020.
  24. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561, 2017.
  25. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Journal of Computational Physics, 404:109136, 2020.
  26. Deepxde: A deep learning library for solving differential equations. SIAM Review, 63(1):208–228, 2021.
  27. A hybrid physics-informed neural network for nonlinear partial differential equation. arXiv preprint arXiv:2112.01696, 2021.
  28. Discontinuity computing with physics-informed neural network. arXiv preprint arXiv:2206.03864, 2022.
  29. Physics-informed neural networks with adaptive localized artificial viscosity. arXiv preprint arXiv:2203.08802, 2022.
  30. wpinns: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws. arXiv preprint arXiv:2207.08483, 2022.
  31. Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the buckley–leverett problem. Scientific reports, 12(1):1–12, 2022.
  32. RoeNets: predicting discontinuity of hyperbolic systems from continuous data. arXiv preprint arXiv:2006.04180, 2020.
  33. A deep learning based discontinuous galerkin method for hyperbolic equations with discontinuous solutions and random uncertainties. arXiv preprint arXiv:2107.01127, 2021.
  34. Triangular mesh methods for the neutron transport equation. Technical report, Los Alamos Scientific Lab., N. Mex.(USA), 1973.
  35. Total variation diminishing runge-kutta schemes. Mathematics of computation, 67(221):73–85, 1998.
  36. Strong stability-preserving high-order time discretization methods. SIAM review, 43(1):89–112, 2001.
  37. Neufenet: Neural finite element solutions with theoretical bounds for parametric pdes. arXiv preprint arXiv:2110.01601, 2021.
  38. Residual dense network for image super-resolution. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2472–2481, 2018.
  39. Rajat Arora. Machine learning-accelerated computational solid mechanics: Application to linear elasticity. arXiv preprint arXiv:2112.08676, 2021.
  40. Rajat Arora. PhySRNet: Physics informed super-resolution network for application in computational solid mechanics. arXiv preprint arXiv:2206.15457, 2022.
  41. Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv preprint arXiv:2001.04536, 2020.
  42. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics, 227(6):3191–3211, 2008.
  43. Phygeonet: Physics-informed geometry-adaptive convolutional neural networks for solving parametric pdes on irregular domain. arXiv e-prints, pages arXiv–2004, 2020.
  44. Dislocation pattern formation in finite deformation crystal plasticity. International Journal of Solids and Structures, 184:114–135, 2020.
  45. A unification of finite deformation J2 Von-Mises plasticity and quantitative dislocation mechanics. Journal of the Mechanics and Physics of Solids, 143:104050, 2020.
  46. Finite element approximation of finite deformation dislocation mechanics. Computer Methods in Applied Mechanics and Engineering, 367:113076, 2020.
  47. Mechanics of micropillar confined thin film plasticity. arXiv preprint arXiv:2202.06410, 2022.
  48. Equilibrium shape of misfitting precipitates with anisotropic elasticity and anisotropic interfacial energy. Modelling and Simulation in Materials Science and Engineering, 28(7):075009, 2020.
  49. A proposal for deployment of wireless sensor network in day-to-day home and industrial appliances for a greener environment. In Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, pages 1081–1086. Springer, 2014.
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