  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) \cite{Weinan2017The} for second order elliptic equations
with Drichilet, Neumann and Robin boundary condition, respectively. We
establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using
deep 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 number
of training samples.
