In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{ReLU}^{3}$ networks, the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{3}$ network, which is of immense independent interest.

翻译：近年来,实际知情的神经网络(PINNs)被证明是实证解决PDEs的有力工具。然而,对PINNs的数值分析仍然缺失。在本文中,我们通过确定训练样本数量、深度和深度的深神经网络的上限,以达到预期准确度,证明Drichlet边界条件的第二顺序椭圆方程式与PINNs的趋同率。PINNs的错误被分解成近似误差和统计误差,其中近似误差以$\mathrm{ReLU}$网络的$2美元标准给出,统计误差由Rademacher复杂度估算。我们用极独立感兴趣的非Lipschitz网络的Rademacher复杂度来推断。