When dealing with a large number of points was required, the traditional 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 purposes 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 problems. Furthermore, rigorous mathematical proofs are provided for our algorithm, demonstrating its validity. Lastly, in the experimental section, 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：当需要处理大量点时，传统的蒙特卡洛方法均匀采样逼近积分的方式效率低下。本文利用数论方法中的好格点集进行采样，并开发了一个深度学习框架，该框架将好格点集与物理信息神经网络相结合。此框架旨在解决低正则性和高维问题。此外，我们为算法提供了严格的数学证明，论证了其有效性。最后，在实验部分，我们通过涉及低正则性泊松方程、二维逆亥姆霍兹方程以及高维线性和非线性问题的数值实验，从数值角度阐明了算法的有效性。