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Convergence analysis of deep Ritz method with over-parameterization.

Zhao Ding1, Yuling Jiao2, Xiliang Lu3

  • 1Wuhan University, School of Mathematics and Statistics, Wuhan, 430072, Asia, China.

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|January 10, 2025
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Summary
This summary is machine-generated.

This study provides the first convergence analysis for the over-parameterized deep Ritz method (DRM) in solving elliptic PDEs. The findings show that network weight norms control convergence rates, irrespective of parameter count, offering insights into deep learning for scientific computing.

Keywords:
Convergence rateDeep Ritz methodDeep approximation with norm controlOver-parameterization

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Area of Science:

  • Numerical Analysis
  • Scientific Computing
  • Machine Learning

Background:

  • The deep Ritz method (DRM) is a promising approach for solving partial differential equations (PDEs).
  • The theoretical underpinnings of why over-parameterized DRM models succeed in numerical analysis remain largely unexplored.
  • Existing numerical analysis for DRM is incomplete, particularly concerning over-parameterization.

Purpose of the Study:

  • To present the first convergence analysis of the over-parameterized deep Ritz method (DRM).
  • To investigate the behavior of DRM for second-order elliptic equations with Robin boundary conditions.
  • To understand the factors influencing the convergence rate in over-parameterized DRM.

Main Methods:

  • Developed a novel convergence analysis framework for over-parameterized DRM.
  • Applied the method to second-order elliptic equations with Robin boundary conditions.
  • Established new approximation results in Sobolev spaces with specific norm constraints.

Main Results:

  • Demonstrated that the convergence rate of DRM is controllable by the network's weight norm.
  • Showed that this control is independent of the total number of parameters in the neural network.
  • The established approximation results in Sobolev spaces have independent theoretical significance.

Conclusions:

  • The over-parameterized deep Ritz method's convergence can be effectively managed through weight norm control.
  • This research provides crucial theoretical justification for the efficacy of over-parameterized deep learning models in solving PDEs.
  • The findings advance the understanding of deep learning applications in scientific computing and numerical analysis.