In this paper, we present refined generalization bounds for the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). For the DRM, we focus on two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube with the Neumann boundary condition. And sharper generalization bounds are derived based on the localization techniques under the assumptions that the exact solutions of the PDEs lie in the Barron spaces or the general Sobolev spaces. For the PINNs, we investigate the general linear second elliptic PDEs with Dirichlet boundary condition via the local Rademacher complexity in the multi-task learning.

翻译：本文针对深度Ritz方法（DRM）与物理信息神经网络（PINNs）提出了精细化的泛化界。对于DRM，我们重点研究了$d$维单位超立方体上具有Neumann边界条件的两种典型椭圆型偏微分方程：泊松方程与静态薛定谔方程。在假设偏微分方程精确解位于Barron空间或一般Sobolev空间的前提下，基于局部化技术推导出了更尖锐的泛化界。对于PINNs，我们借助多任务学习中的局部Rademacher复杂度，研究了具有Dirichlet边界条件的一般线性二阶椭圆型偏微分方程。