Analysis of Deep Ritz Methods for Semilinear Elliptic Equations  

作　　者：Mo Chen Yuling Jiao Xiliang Lu Pengcheng Song Fengru Wang Jerry Zhijian Yang 

机构地区：[1]School of Mathematics and Statistics,Wuhan University,299 Ba Yi Road,Wuhan 430072,P.R.China [2]Hubei Key Laboratory of Computational Science,Wuhan University,299 Ba Yi Road,Wuhan 430072,P.R.China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2024年第1期181-209,共29页高等学校计算数学学报（英文版）

基　　金：supported by the National Key Research and Development Program of China(Grant No.2020YFA0714200);the National Nature Science Foundation of China(Grant Nos.12125103,12071362,12371424,12371441);supported by the Fundamental Research Funds for the Central Universities.The numerical calculations have been done at the Supercomputing Center of Wuhan University.

摘　　要：In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 activations.Firstly,we present a comprehensive formulation based on the penalized variational form of the elliptical equations.We then apply the Deep Ritz Method,which works for a wide range of equations.We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth D,width W of the ReLU2 ResNet,and the num-ber of training samples n.Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

关 键 词：Semilinear elliptic equations Deep Ritz method ReLUMo ResNet convergence rate 

分 类 号：O17[理学—数学]