A novel number-theoretic sampling method for neural network solutions of partial differential equations

TL;DR

This paper introduces a new deep learning sampling method based on number theory that improves the efficiency and accuracy of neural network solutions for complex, high-dimensional partial differential equations, outperforming traditional Monte Carlo approaches.

Contribution

The paper presents a novel deterministic sampling framework using number-theoretic points integrated with PINNs, providing mathematical guarantees and superior performance in solving challenging PDEs.

Findings

Lower error bounds compared to uniform sampling

Effective in high-dimensional PDEs

Demonstrated superior accuracy in numerical experiments

Abstract

Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling points, which is a robust approach specifically designed to handle partial differential equations with rough solutions or in high dimensions. The sample points are generated by the generating vector to achieve the smallest discrepancy. The architecture integrates Physics-Informed Neural Networks (PINNs) with rigorous mathematical guarantees demonstrating lower error bounds compared to conventional uniform random sampling. Numerical validation includes low-regularity Poisson equations, two-dimensional inverse Helmholtz problems, and high-dimensional linear/nonlinear PDEs, systematically demonstrating the algorithm's superior performance and generalization capabilities.