Analysis of Deep Ritz Methods for Semilinear Elliptic Equations
Year: 2024
Author: Mo Chen, Yuling Jiao, Xiliang Lu, Pengcheng Song, Fengru Wang, Jerry Zhijian Yang
Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 1 : pp. 181–209
Abstract
In this paper, we propose a method for solving semilinear elliptical equations 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 number of training samples n. Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2023-0058
Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 1 : pp. 181–209
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Semilinear elliptic equations Deep Ritz method ReLU2 ResNet convergence rate.
Author Details
Mo Chen Email
Yuling Jiao Email
Xiliang Lu Email
Pengcheng Song Email
Fengru Wang Email
Jerry Zhijian Yang Email