Stable Weight Updating: A Key to Reliable PDE Solutions Using Deep Learning

TL;DR

This paper introduces residual-based neural network architectures to improve stability and accuracy in solving complex PDEs with deep learning, demonstrating their effectiveness through extensive experiments.

Contribution

The paper presents novel residual-based architectures, the Simple Highway Network and Squared Residual Network, enhancing stability and accuracy in physics-informed neural networks for PDE solutions.

Findings

Squared Residual Network shows superior stability and accuracy.

Residual architectures outperform traditional neural networks in PDE solving.

Enhanced robustness in nonlinear and time-dependent PDEs.

Abstract

Background: Deep learning techniques, particularly neural networks, have revolutionized computational physics, offering powerful tools for solving complex partial differential equations (PDEs). However, ensuring stability and efficiency remains a challenge, especially in scenarios involving nonlinear and time-dependent equations. Methodology: This paper introduces novel residual-based architectures, namely the Simple Highway Network and the Squared Residual Network, designed to enhance stability and accuracy in physics-informed neural networks (PINNs). These architectures augment traditional neural networks by incorporating residual connections, which facilitate smoother weight updates and improve backpropagation efficiency. Results: Through extensive numerical experiments across various examples including linear and nonlinear, time-dependent and independent PDEs we demonstrate the efficacy of the proposed architectures. The Squared Residual Network, in particular, exhibits robust performance, achieving enhanced stability and accuracy compared to conventional neural networks. These findings underscore the potential of residual-based architectures in advancing deep learning for PDEs and computational physics applications.