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Volume 33, Issue 2
Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

Sidi Wu, Aiqing Zhu, Yifa Tang & Benzhuo Lu

DOI: 10.4208/cicp.OA-2022-0218

Commun. Comput. Phys., 33 (2023), pp. 596-627.

Published online: 2023-03

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  • Abstract

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.

  • Keywords

Elliptic interface problems, generalization errors, convergence analysis, neural networks.

  • AMS Subject Headings

65N12, 82B24, 68T07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-33-596, author = {Wu , SidiZhu , AiqingTang , Yifa and Lu , Benzhuo}, title = {Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {2}, pages = {596--627}, abstract = {

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0218}, url = {http://global-sci.org/intro/article_detail/cicp/21501.html} }
TY - JOUR T1 - Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems AU - Wu , Sidi AU - Zhu , Aiqing AU - Tang , Yifa AU - Lu , Benzhuo JO - Communications in Computational Physics VL - 2 SP - 596 EP - 627 PY - 2023 DA - 2023/03 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0218 UR - https://global-sci.org/intro/article_detail/cicp/21501.html KW - Elliptic interface problems, generalization errors, convergence analysis, neural networks. AB -

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.

Sidi Wu, Aiqing Zhu, Yifa Tang & Benzhuo Lu. (2023). Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems. Communications in Computational Physics. 33 (2). 596-627. doi:10.4208/cicp.OA-2022-0218
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