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

2026-03-03

AI Summary

This work addresses the challenges of training imbalance and solution inaccuracy that conventional physics-informed neural networks (PINNs) face when solving high-stiffness or shock-dominated partial differential equations. To overcome these limitations, the authors propose a novel framework integrating an adaptive loss weighting mechanism based on smoothed gradient norms with a residual-driven dynamic collocation point strategy. This approach significantly improves the satisfaction of initial and boundary conditions while enhancing solution accuracy in regions of high residual. The method demonstrates markedly improved stability and precision for complex dynamical systems, achieving relative L2 error reductions of 44% and 70% on the Burgers and Allen–Cahn equations, respectively, with results closely matching those obtained by finite difference methods.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Natural Language Processing: Safety and Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Physics-Informed Neural Networks
• unbalanced training
• solution inaccuracy
• high stiffness
• shock-dominated dynamics

Innovation

• adaptive loss balancing
• residual-based collocation
• physics-informed neural networks
• high stiffness dynamics
• smoothed gradient norms