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.

翻译：本文分别分析了在$\mathbb{R}^d$上具有Dirichlet、Neumann和Robin边界条件的二阶椭圆偏微分方程的深层伽辽金弱解方法（DGMW）的收敛速率。在DGMW中，采用深度神经网络参数化PDE解，并采用第二个神经网络参数化传统伽辽金公式中的检验函数。通过根据训练样本数量$n$适当选择这两个网络的深度和宽度，证明了DGMW的收敛速率为$\mathcal{O}(n^{-1/d})$，这是针对弱解的首个收敛性结果。证明的主要思想是将DGMW的误差划分为逼近误差和统计误差。我们推导了$H^{1}$范数下逼近误差的上界，并通过Rademacher复杂度界定了统计误差。