Moving sample method for solving time-dependent partial differential equations

TL;DR

This paper introduces an adaptive sampling method for physics-informed neural networks to efficiently solve time-dependent PDEs with sharp gradients, focusing computational resources on challenging regions for improved accuracy.

Contribution

It proposes a residual-driven adaptive sampling framework that iteratively updates training point distribution based on error estimates, enhancing efficiency and solution quality.

Findings

Achieves higher accuracy with fewer points than uniform sampling.

Effectively concentrates resources on regions with significant residuals.

Improves solution quality on benchmark PDE problems.

Abstract

Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point allocation that wastes resources on regions already well-resolved. This paper presents an adaptive sampling framework for PINNs aimed at efficiently solving time-dependent partial differential equations with pronounced local singularities. The method employs a residual-driven strategy, where the spatial-temporal distribution of training points is iteratively updated according to the error field from the previous iteration. This targeted allocation enables the network to concentrate computational effort on regions with significant residuals, achieving higher accuracy with fewer sampling points compared to uniform sampling. Numerical experiments on representative PDE benchmarks demonstrate that the proposed approach improves solution quality.