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：本文提出了针对深度Ritz方法（DRM）和物理信息神经网络（PINNs）的精细化泛化界。对于DRM，我们重点研究两类原型椭圆偏微分方程：在d维单位超立方体上具有诺伊曼边界条件的泊松方程和静态薛定谔方程。在精确解属于Barron空间或一般Sobolev空间的假设下，基于局部化技术推导出更尖锐的泛化界。对于PINNs，我们通过多任务学习中的局部Rademacher复杂度，研究了具有狄利克雷边界条件的一般线性二阶椭圆偏微分方程。