Self-adaptive weighting and sampling for physics-informed neural networks

TL;DR

This paper presents a hybrid adaptive sampling and weighting approach to improve the accuracy and efficiency of physics-informed neural networks in solving complex PDEs, especially with limited training data.

Contribution

The paper introduces a combined adaptive sampling and weighting framework that enhances PINNs' performance over existing methods, addressing challenges in training on complex problems.

Findings

Combined approach improves prediction accuracy

Enhances training efficiency in complex PDEs

More robust performance with limited data

Abstract

Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and efficiency. In this work, we introduce a hybrid adaptive sampling and weighting method to enhance the performance of physics-informed neural networks (PINNs). The adaptive sampling component identifies training points in regions where the solution exhibits rapid variation, while the adaptive weighting component balances the convergence rate across training points. Numerical experiments show that applying only adaptive sampling or only adaptive weighting is insufficient to consistently achieve accurate predictions, particularly when training points are scarce. Since each method emphasizes different aspects of the solution, their effectiveness is problem dependent. By combining both strategies, the proposed framework consistently improves prediction accuracy and training efficiency, offering a more robust approach for solving PDEs with PINNs.