Volume 28, Issue 5
On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs

Commun. Comput. Phys., 28 (2020), pp. 2042-2074.

Published online: 2020-11

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• Abstract

Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.
As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.

• Keywords

Physics informed neural networks, convergence, Hölder regularization, elliptic and parabolic PDEs, Schauder approach.

65M12, 41A46, 35J25, 35K20

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@Article{CiCP-28-2042, author = {Shin , Yeonjong and Darbon , Jérôme and Em Karniadakis , George}, title = {On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {5}, pages = {2042--2074}, abstract = {

Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.
As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0193}, url = {http://global-sci.org/intro/article_detail/cicp/18404.html} }
TY - JOUR T1 - On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs AU - Shin , Yeonjong AU - Darbon , Jérôme AU - Em Karniadakis , George JO - Communications in Computational Physics VL - 5 SP - 2042 EP - 2074 PY - 2020 DA - 2020/11 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2020-0193 UR - https://global-sci.org/intro/article_detail/cicp/18404.html KW - Physics informed neural networks, convergence, Hölder regularization, elliptic and parabolic PDEs, Schauder approach. AB -

Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs.
As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.

Yeonjong Shin, Jérôme Darbon & George Em Karniadakis. (2020). On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs. Communications in Computational Physics. 28 (5). 2042-2074. doi:10.4208/cicp.OA-2020-0193
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