Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

A. Noorizadegan, R. Cavoretto, D. L. Young, C. S. Chen

TL;DR

This work tackles stability challenges in physics-informed neural networks (PINNs) used to solve partial differential equations. It introduces two residual-based architectures, the Simple Highway Network (Simple HwNet) and the Squared Residual Network (Sqr-ResNet), designed to stabilize weight updates and enhance convergence. Across linear and nonlinear, time-dependent and independent PDEs, Sqr-ResNet demonstrates superior stability and accuracy, with Simple HwNet offering notable improvements over plain networks. The findings suggest that residual-based architectural choices can significantly advance robust PDE solving with deep learning, broadening the practical impact of PINNs in computational physics.

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.

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

TL;DR

This work tackles stability challenges in physics-informed neural networks (PINNs) used to solve partial differential equations. It introduces two residual-based architectures, the Simple Highway Network (Simple HwNet) and the Squared Residual Network (Sqr-ResNet), designed to stabilize weight updates and enhance convergence. Across linear and nonlinear, time-dependent and independent PDEs, Sqr-ResNet demonstrates superior stability and accuracy, with Simple HwNet offering notable improvements over plain networks. The findings suggest that residual-based architectural choices can significantly advance robust PDE solving with deep learning, broadening the practical impact of PINNs in computational physics.

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.
Paper Structure (15 sections, 48 equations, 8 figures, 5 tables)

This paper contains 15 sections, 48 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Plain Neural Networks
  3. Multi-Layer Perceptrons (MLPs)
  4. Layer Configuration
  5. Network Parameterization
  6. Back-propagation for Gradient Computation
  7. Training and Testing of MLPs
  8. Overview of Physics-Informed Neural Networks
  9. Advanced Deep Neural Network Architectures
  10. Highway Networks
  11. Residual Networks
  12. A Simple Highway Network (Simple HwNet)
  13. Proposed Squared Residual Network (Sqr-ResNet)
  14. Results and Discussions
  15. Conclusions

Figures (8)

  • Figure 1: Backpropagation expressions for a 2-layer network.
  • Figure 2: four neural network architectures: (a) a basic neural network (Plain Net), (b) Highway network (Hw), and (c) ResNet (d) the proposed simple highway network (simple HwNet) (e) Square-Residual network (Sqr-ResNet).
  • Figure 3: Example 1: Validation error (L2 norm error) for three networks.
  • Figure 4: Example 1: Approximated function and norm of the error for three networks. Left panel: PINN; middle panel: Simple highway network; right panel: square ResNet.
  • Figure 5: Example 1: The histogram of gradient of loss function with respect to the weights for the plain network and proposed square-ResNet.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5