Convergence analysis of deep Ritz method with over-parameterization.

Journal: Neural networks : the official journal of the International Neural Network Society
Published Date: Jan 4, 2025

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.

Authors

  • Zhao Ding
    Institute of Basic Medical Sciences of Xiyuan Hospital, China Academy of Chinese Medical Sciences, Beijing Key Laboratory of Chinese Materia Pharmacology, National Clinical Research Center of Traditional Chinese Medicine for Cardiovascular Diseases, Beijing, China.
  • Yuling Jiao
    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.
  • Xiliang Lu
    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.
  • Peiying Wu
    Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, Asia, China. Electronic address: peiyingwu@whu.edu.cn.
  • Jerry Zhijian Yang
    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.

Keywords

External Resources

Popular Topics
Recent Journals