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 space or the general Sobolev spaces. For the PINNs, we investigate the general linear second elliptic PDEs with Dirichlet boundary condition.

翻译：本文对深度Ritz方法（DRM）和物理信息神经网络（PINNs）提出了更精细的泛化界。针对DRM，我们聚焦于两类典型椭圆型偏微分方程：在d维单位超立方体上具有纽曼边界条件的泊松方程和静态薛定谔方程。基于局部化技术，在PDE精确解属于Barron空间或一般索伯列夫空间的假设下，推导出更精确的泛化界。针对PINNs，我们研究了具有狄利克雷边界条件的一般线性二阶椭圆型偏微分方程。