Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Condition (CSIAM Transactions on Applied Mathematics, 2022)
Deep Ritz methods (DRM) have been proven numerically to be efficient insolving partial differential equations. In this paper, we present a convergence rate in $H^1$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks andthe number of samples explicitly. Further we can properly choose the depth and widthin the deep neural networks in terms of the number of training samples. The main ideaof the proof is to decompose the total error of DRM into three parts, that is approximation error, statistical error and the error caused by the boundary penalty. We bound theapproximation error in $H^1$ norm with ${\rm ReLU}^2$ networks and control the statistical errorvia Rademacher complexity. In particular, we derive the bound on the Rademachercomplexity of the non-Lipschitz composition of gradient norm with ${\rm ReLU}^2$ network,which is of immense independent interest. We also analyze the error inducing by theboundary penalty method and give a prior rule for tuning the penalty parameter.
Keywords: Deep Ritz methods; convergence rate; Dirichlet boundary condition; approximation error; Rademacher complexity.