Quasi-Random Physics-informed Neural Networks

TL;DR

This paper introduces Quasi-Random Physics-Informed Neural Networks (QRPINNs), which utilize low-discrepancy sequences for sampling, leading to improved convergence and performance over traditional PINNs, especially in high-dimensional PDE problems.

Contribution

The paper proposes QRPINNs that incorporate quasi-Monte Carlo sampling, providing theoretical convergence guarantees and empirical performance improvements over existing PINNs.

Findings

QRPINNs have better convergence rates than PINNs.

QRPINNs outperform PINNs and adaptive sampling methods in high-dimensional PDEs.

Combining QRPINNs with adaptive sampling further enhances performance.

Abstract

Physics-informed neural networks have shown promise in solving partial differential equations (PDEs) by integrating physical constraints into neural network training, but their performance is sensitive to the sampling of points. Based on the impressive performance of quasi Monte-Carlo methods in high dimensional problems, this paper proposes Quasi-Random Physics-Informed Neural Networks (QRPINNs), which use low-discrepancy sequences for sampling instead of random points directly from the domain. Theoretically, QRPINNs have been proven to have a better convergence rate than PINNs. Empirically, experiments demonstrate that QRPINNs significantly outperform PINNs and some representative adaptive sampling methods, especially in high-dimensional PDEs. Furthermore, combining QRPINNs with adaptive sampling can further improve the performance.