Stable Weight Updating: A Key to Reliable PDE Solutions Using Deep Learning
This addresses stability issues in physics-informed neural networks for computational physics, though it appears incremental as it builds on existing PINN methods with architectural modifications.
The paper tackled the challenge of ensuring stability and efficiency in solving partial differential equations (PDEs) using deep learning, by introducing residual-based architectures like the Squared Residual Network, which demonstrated enhanced stability and accuracy compared to conventional neural networks in numerical experiments.
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.