Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems
Year: 2023
Author: Sidi Wu, Aiqing Zhu, Yifa Tang, Benzhuo Lu
Communications in Computational Physics, Vol. 33 (2023), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2022-0218
Communications in Computational Physics, Vol. 33 (2023), Iss. 2 : pp. 596–627
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Elliptic interface problems generalization errors convergence analysis neural networks.
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