Number Theoretic Accelerated Learning of Physics-Informed Neural Networks

TL;DR

This paper introduces number theoretic techniques to improve the efficiency of physics-informed neural networks by reducing the number of collocation points needed, thus lowering computational costs while maintaining performance.

Contribution

It proposes good lattice training and periodization tricks inspired by number theory to accelerate PDE solving with neural networks.

Findings

GLT requires 2-7 times fewer collocation points

Lower computational cost achieved

Maintains competitive performance

Abstract

Physics-informed neural networks solve partial differential equations by training neural networks. Since this method approximates infinite-dimensional PDE solutions with finite collocation points, minimizing discretization errors by selecting suitable points is essential for accelerating the learning process. Inspired by number theoretic methods for numerical analysis, we introduce good lattice training and periodization tricks, which ensure the conditions required by the theory. Our experiments demonstrate that GLT requires 2-7 times fewer collocation points, resulting in lower computational cost, while achieving competitive performance compared to typical sampling methods.