Convergence Analysis of the Deep Galerkin Method for Weak Solutions

TL;DR

This paper establishes the first convergence rate for the deep Galerkin method applied to weak solutions of second-order elliptic PDEs, showing it converges at a rate of (n^{-1/d}) with neural network parametrization.

Contribution

It provides the first theoretical convergence analysis for DGMW applied to weak solutions, linking network architecture to convergence rate.

Findings

Convergence rate of (n^{-1/d}) for DGMW.

Error decomposition into approximation and statistical errors.

Upper bounds on errors using Rademacher complexity.

Abstract

This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on $\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions, respectively. In DGMW, a deep neural network is applied to parametrize the PDE solution, and a second neural network is adopted to parametrize the test function in the traditional Galerkin formulation. By properly choosing the depth and width of these two networks in terms of the number of training samples $n$, it is shown that the convergence rate of DGMW is $\mathcal{O}(n^{-1/d})$, which is the first convergence result for weak solutions. The main idea of the proof is to divide the error of the DGMW into an approximation error and a statistical error. We derive an upper bound on the approximation error in the $H^{1}$ norm and bound the statistical error via Rademacher complexity.