Error analysis of deep Ritz methods for elliptic equations

Abstract

Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention 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 the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann and Robin boundary conditions, respectively. We establish the first nonasymptotic convergence rate in H1 norm for DRM using deep neural 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 the number of training samples.