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Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

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