In this letter, we investigate the statistical limits of deep learning for learning solutions of elliptic interface problems from randomly generated data by employing Physics-informed Neural Networks (PINNs). We prove a lower bound and an upper bound for the generalization error of PINNs for solving elliptic interface equations with the Dirichlet boundary condition. In particular, the upper bound on the generalization error of PINNs is obtained by computing the fixed points of the local Rademacher complexity. We show that variations in volume across distinct regions exert influence on the requisite sample complexity. This unveils an underlying principle within the sampling strategy: the sample size of the data is contingent upon the Lebesgue measure of the domain, coupled with the smoothness and dimensionality characteristics of the interface problem. This letter sheds light on the network architecture design for solving interface problems using PINNs.

中文翻译：

---

椭圆界面问题的 PINN 推广


在这封信中，我们研究了深度学习的统计限制，用于通过采用物理信息神经网络（PINN）从随机生成的数据中学习椭圆界面问题的解决方案。我们证明了用 Dirichlet 边界条件求解椭圆界面方程的 PINN 泛化误差的下界和上限。特别是，PINN 泛化误差的上限是通过计算局部 Rademacher 复杂度的不动点来获得的。我们表明，不同区域的体积变化会对所需的样本复杂性产生影响。这揭示了采样策略中的一个基本原则：数据的样本大小取决于域的勒贝格测度，以及界面问题的平滑度和维度特征。这封信阐明了使用 PINN 解决接口问题的网络架构设计。