Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective

TL;DR

This paper introduces an adaptive weighting scheme for physics-informed neural networks (PINNs) that improves convergence and accuracy by focusing on continuous neighborhood regions rather than isolated points, supported by empirical error reduction.

Contribution

It proposes a novel primal-dual optimization-based weighting method for PINNs that adaptively emphasizes neighborhood regions to enhance solution accuracy.

Findings

Reduces relative L2 errors in PINNs

Improves convergence of PINN training

Enhances accuracy by focusing on neighborhood regions

Abstract

Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by computational limitations, PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy. In this study, we proposed a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions. The empirical results show that our weighting scheme can reduce the relative $L^2$ errors to a lower value.