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A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs

A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs
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Year:    2022

Author:    Yuling Jiao, Yanming Lai, Dingwei Li, Xiliang Lu, Fengru Wang, Yang Wang, Jerry Zhijian Yang

Communications in Computational Physics, Vol. 31 (2022), Iss. 4 : pp. 1272–1295

Abstract

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU$^3$ network, which is of immense independent interest.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2021-0186

Communications in Computational Physics, Vol. 31 (2022), Iss. 4 : pp. 1272–1295

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    PINNs ReLU$^3$ neural network B-splines Rademacher complexity

Author Details

Yuling Jiao

Yanming Lai

Dingwei Li

Xiliang Lu

Fengru Wang

Yang Wang

Jerry Zhijian Yang

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