Convergence Rate Analysis for Deep Ritz Method

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the learning method works is falling far behind its empirical success. In this paper, we provide rigorous numerical analysis on Ritz (DRM) \cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish first nonasymptotic convergence rate in $H^1$ norm DRM using $\mathrm{ReLU}^2$ activation functions. addition providing theoretical justification DRM, our study also shed light how set hyper-parameter depth and width achieve desired terms number training samples. Technically, derive bounds approximation error network Rademacher complexity non-Lipschitz composition gradient network, both which are independent interest.