Error Analysis of Deep Ritz Methods for Elliptic Equations

TL;DR

This paper provides a rigorous convergence analysis of deep Ritz methods for elliptic PDEs, establishing nonasymptotic rates and guidelines for hyper-parameter tuning to improve theoretical understanding of deep learning solutions for PDEs.

Contribution

It offers the first nonasymptotic convergence rate analysis for deep Ritz methods with smooth activation functions in solving elliptic equations.

Findings

Established convergence rates in $H^1$ norm.

Guidelines for setting depth and width of neural networks.

Analysis covers Dirichlet, Neumann, and Robin boundary conditions.

Abstract

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{Weinan2017The} for second order elliptic equations with Drichilet, Neumann and Robin boundary condition, respectively. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples.