Convergence Rate Analysis for Deep Ritz Method

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) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bound on the approximation error of deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are of independent interest.