On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs

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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.
Publisher
GLOBAL SCIENCE PRESS
Issue Date
2020-11
Language
English
Article Type
Article
Citation

COMMUNICATIONS IN COMPUTATIONAL PHYSICS, v.28, no.5, pp.2042 - 2074

ISSN
1815-2406
DOI
10.4208/cicp.OA-2020-0193
URI
http://hdl.handle.net/10203/297249
Appears in Collection
MA-Journal Papers(저널논문)
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