Stabilized Adaptive Loss and Residual-Based Collocation for Physics-Informed Neural Networks

TL;DR

This paper introduces adaptive loss balancing and residual-based collocation techniques to enhance the accuracy and stability of Physics-Informed Neural Networks when solving stiff and shock-dominated PDEs.

Contribution

The paper proposes novel adaptive schemes for loss balancing and residual-based collocation to improve PINNs' performance on challenging PDEs with high stiffness or shocks.

Findings

44% reduction in L2 error for Burgers' equation

70% reduction in L2 error for Allen-Cahn equation

Improved solution accuracy and physics residual satisfaction

Abstract

Physics-Informed Neural Networks (PINNs) have been recognized as a mesh-free alternative to solve partial differential equations where physics information is incorporated. However, in dealing with problems characterized by high stiffness or shock-dominated dynamics, traditional PINNs have been found to have limitations, including unbalanced training and inaccuracy in solution, even with small physics residuals. In this research, we seek to address these limitations using the viscous Burgers' equation with low viscosity and the Allen-Cahn equation as test problems. In addressing unbalanced training, we have developed a new adaptive loss balancing scheme using smoothed gradient norms to ensure satisfaction of initial and boundary conditions. Further, to address inaccuracy in the solution, we have developed an adaptive residual-based collocation scheme to improve the accuracy of solutions in the regions with high physics residuals. The proposed new approach significantly improves solution accuracy with consistent satisfaction of physics residuals. For instance, in the case of Burgers' equation, the relative L2 error is reduced by about 44 percent compared to traditional PINNs, while for the Allen-Cahn equation, the relative L2 error is reduced by approximately 70 percent. Additionally, we show the trustworthy solution comparison of the proposed method using a robust finite difference solver.