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Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations (2404.15618v1)
Published 24 Apr 2024 in stat.ML and cs.LG
Abstract: The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the capability to provide uncertainty measures for their predictions, a crucial aspect, especially in data-driven scenarios with limited available data. In this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP), which exploits the probabilistic characteristics of Gaussian Processes (GPs) while leveraging the learning prowess of operator learning. The proposed framework leads to improved prediction accuracy and offers a quantifiable measure of uncertainty. The proposed framework is extensively evaluated through experiments on various PDE examples, including Burger's equation, Darcy flow, non-homogeneous Poisson, and wave-advection equations. Furthermore, a comparative study with state-of-the-art operator learning algorithms is presented to highlight the advantages of NOGaP. The results demonstrate superior accuracy and expected uncertainty characteristics, suggesting the promising potential of the proposed framework.
- Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020.
- Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193, 2019.
- Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems. Computer Methods in Applied Mechanics and Engineering, 404:115783, 2023.
- Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators, 2022.
- Spherical neural operator network for global weather prediction. IEEE Transactions on Circuits and Systems for Video Technology, 2023.
- A physics-informed variational deeponet for predicting crack path in quasi-brittle materials. Computer Methods in Applied Mechanics and Engineering, 391:114587, March 2022.
- Fault detection and isolation using probabilistic wavelet neural operator auto-encoder with application to dynamic processes. Process Safety and Environmental Protection, 173:215–228, 2023.
- Vb-deeponet: A bayesian operator learning framework for uncertainty quantification. Engineering Applications of Artificial Intelligence, 118:105685, 2023.
- Approximate bayesian neural operators: Uncertainty quantification for parametric pdes. ArXiv, abs/2208.01565, 2022.
- Solving and learning nonlinear pdes with gaussian processes, 2021.
- Kernel methods are competitive for operator learning, 2023.
- Sparse cholesky factorization for solving nonlinear pdes via gaussian processes. ArXiv, abs/2304.01294, 2023.
- Physics informed wno, 2023.
- Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3):218–229, 2021.
- Learning the solution operator of parametric partial differential equations with physics-informed deeponets. Science advances, 7(40):eabi8605, 2021.
- Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363:55–78, 2018.
- Model reduction and neural networks for parametric pdes. The SMAI journal of computational mathematics, 7:121–157, 2021.
- Neural operator: Graph kernel network for partial differential equations, 2020.
- Fourier neural operator with learned deformations for pdes on general geometries, 2022.
- Geometry-informed neural operator for large-scale 3d pdes. Advances in Neural Information Processing Systems, 36, 2024.
- A foundational neural operator that continuously learns without forgetting, 2023.
- Multi-fidelity wavelet neural operator with application to uncertainty quantification, 2023.
- Generative adversarial wavelet neural operator: Application to fault detection and isolation of multivariate time series data, 2024.
- N Navaneeth and Souvik Chakraborty. Waveformer for modelling dynamical systems, 2023.
- Markus Lange-Hegermann. Algorithmic linearly constrained gaussian processes, 2019.
- A survey of constrained gaussian process regression: Approaches and implementation challenges. Journal of Machine Learning for Modeling and Computing, 1(2), 2020.
- Gaussian process regression constrained by boundary value problems. Computer Methods in Applied Mechanics and Engineering, 388:114117, 2022.
- B-pinns: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data. Journal of Computational Physics, 425:109913, January 2021.
- Variational bayes deep operator network: A data-driven bayesian solver for parametric differential equations, 2022.
- Neuraluq: A comprehensive library for uncertainty quantification in neural differential equations and operators, 2022.
- Physics-informed gaussian process regression generalizes linear pde solvers, 2023.
- Sparse cholesky factorization for solving nonlinear pdes via gaussian processes, 2023.
- Constraining gaussian processes to systems of linear ordinary differential equations, 2022.
- Multi-output separable gaussian process: Towards an efficient, fully bayesian paradigm for uncertainty quantification. Journal of Computational Physics, 241:212–239, 2013.
- Multi-task gaussian process prediction. Advances in neural information processing systems, 20, 2007.
- Kernels for vector-valued functions: a review, 2012.
- Kevin P Murphy. Probabilistic machine learning: an introduction. MIT press, 2022.
- Carl Edward Rasmussen. Gaussian Processes in Machine Learning, pages 63–71. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004.
- C Bishop. Pattern recognition and machine learning. Springer google schola, 2:531–537, 2006.
- Gaussian error linear units (gelus), 2023.