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.

翻译：在处理大量采样点时，通常采用均匀采样结合蒙特卡洛方法近似积分的策略会变得低效。本文利用数论方法中的优良格点集进行采样，构建了一个将优良格点集与物理信息神经网络相融合的深度学习框架。该框架旨在求解低正则性及高维偏微分方程。此外，我们提供了严格的数学证明，验证了所提方法的误差上界小于均匀采样方法。通过数值实验，包括低正则性泊松方程、二维逆亥姆霍兹方程以及高维线性和非线性问题，从数值角度验证了本算法的有效性。