Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks

TL;DR

This paper introduces a pointwise adaptive weighting method for physics-informed neural networks that balances residual decay rates, improving accuracy and efficiency in solving complex partial differential equations.

Contribution

The paper proposes a novel adaptive weighting technique based on residual decay rates, enhancing convergence and accuracy over existing methods in physics-informed deep learning.

Findings

Balanced residual decay rates improve convergence speed.

The method achieves higher prediction accuracy with lower computational cost.

It demonstrates robustness and ease of hyperparameter tuning.

Abstract

Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. However, for complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy and efficiency. In this work, we demonstrate that the failure of plain physics-informed neural networks arises from the significant discrepancy in the convergence rate of residuals at different training points, where the slowest convergence rate dominates the overall solution convergence. Based on these observations, we propose a pointwise adaptive weighting method that balances the residual decay rate across different training points. The performance of our proposed adaptive weighting method is compared with current state-of-the-art adaptive weighting methods on benchmark problems for both physics-informed neural networks and physics-informed deep operator networks. Through extensive numerical results we demonstrate that our proposed approach of balanced residual decay rates offers several advantages, including bounded weights, high prediction accuracy, fast convergence rate, low training uncertainty, low computational cost, and ease of hyperparameter tuning.