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

查看全文 作　　者：Yuling [1,2]Jiao;Yanming [1]Lai;Dingwei [1]Li;Xiliang [1,2]Lu;Fengru [1,2]Wang;Yang [3]Wang;Jerry Zhijian [1,2]Yang

机构地区：[1]School of Mathematics and Statistics,Wuhan University,P.R.China;[2]Hubei Key Laboratory of Computational Science,Wuhan University,P.R.China;[3]Department ofMathematics,The Hong Kong University of Science and Technology,Clear Water Bay,Kowloon,Hong Kong

出　　处：《Communications in Computational Physics》索引2022年第31卷第4期,共24页

基　　金：supported by the National Key Research and Development Program of China(No.2020YFA0714200);the National Science Foundation of China(No.12125103,No.12071362,No.11971468,No.11871474,No.11871385);the Natural Science Foundation of Hubei Province(No.2021AAA010,No.2019CFA007);the Fundamental Research Funds for the Central Universities. 摘　　要：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 C2 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. 关键词：PINNs ReLU^(3)neural network B-SPLINES Rademacher complexity

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

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