Convergence Rate Analysis for Deep Ritz Method (Communications in Computational Physics, 2022)
Using deep neural networks to solve PDEs has attracted a lot of attentionsrecently. 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 Ritzmethod (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRMusing deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing atheoretical 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 ofnumber of training samples. Technically, we derive bound on the approximation errorof deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of thenon-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are ofindependent interest.
Keywords: Deep Ritz method; deep neural networks; convergence rate analysis.