A good lattice point sampling neural network for solving partial differential equations

When dealing with a large number of points, the usually uniform sampling approach for approximating integrals using the Monte Carlo method becomes inefficient. In this work, we leverage the good lattice point sets from number-theoretic methods for sampling and develop a deep learning framework that integrates the good lattice point sets with Physics-Informed Neural Networks. This framework is designed to address low-regularity and high-dimensional partial differential equations. Furthermore, rigorous mathematical proofs are provided to validate the error bound of our method is less than that of uniform sampling methods. We employ numerical experiments involving the Poisson equation with low regularity, the two-dimensional inverse Helmholtz equation, and high-dimensional linear and nonlinear problems to illustrate the effectiveness of our algorithm.% from a numerical perspective.