Convergence analysis of deep Ritz method with over-parameterization

Zhao Ding; Yuling Jiao; Xiliang Lu; Peiying Wu; Jerry Zhijian Yang

doi:10.1016/j.neunet.2024.107110

Convergence analysis of deep Ritz method with over-parameterization

Neural Netw. 2025 Jan 4:184:107110. doi: 10.1016/j.neunet.2024.107110. Online ahead of print.

Authors

Zhao Ding¹, Yuling Jiao², Xiliang Lu³, Peiying Wu⁴, Jerry Zhijian Yang⁵

Affiliations

¹ Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, Asia, China. Electronic address: zd1998@whu.edu.cn.
² Wuhan University, National Center for Applied Mathematics in Hubei, Wuhan, 430072, Asia, China; Wuhan University, Hubei Key Laboratory of Computational Science, Wuhan, 430072, Asia, China; Wuhan University, School of Artificial Intelligence, Wuhan, 430072, Asia, China. Electronic address: yulingjiaomath@whu.edu.cn.
³ Wuhan University, National Center for Applied Mathematics in Hubei, Wuhan, 430072, Asia, China; Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, Asia, China; Wuhan University, Hubei Key Laboratory of Computational Science, Wuhan, 430072, Asia, China. Electronic address: xllv.math@whu.edu.cn.
⁴ Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, Asia, China. Electronic address: peiyingwu@whu.edu.cn.
⁵ Wuhan University, National Center for Applied Mathematics in Hubei, Wuhan, 430072, Asia, China; Wuhan University, Wuhan Institute for Math & AI, Wuhan, 430072, Asia, China; Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, Asia, China; Wuhan University, Hubei Key Laboratory of Computational Science, Wuhan, 430072, Asia, China. Electronic address: zjyang.math@whu.edu.cn.

PMID: 39793492
DOI: 10.1016/j.neunet.2024.107110

Abstract

The deep Ritz method (DRM) has recently been shown to be a simple and effective method for solving PDEs. However, the numerical analysis of DRM is still incomplete, especially why over-parameterized DRM works remains unknown. This paper presents the first convergence analysis of the over-parameterized DRM for second-order elliptic equations with Robin boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network. To this end, we establish novel approximation results in Sobolev spaces with norm constraints, which have independent significance.

Keywords: Convergence rate; Deep Ritz method; Deep approximation with norm control; Over-parameterization.