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.

翻译：使用深神经网络解决PDE最近引起了许多关注。 但是, 深学习方法的运行远远落后于其成功的经验。 在本文中, 我们提供了对深Ritz方法( DRM)\ cite{Weinan2017The} 的严格数字分析, 分别用于Drichilet、 Neumann 和 Robin 边界条件的第二顺序椭圆方程。 我们用具有平稳激活功能的深网络, 包括后勤和双曲正切功能, 为DRM 建立了第一个以$H1美元为标准的非自然聚合率。 我们的结果显示如何设定深度和宽度的超分数, 以培训样本的数量达到预期的趋同率 。