Stabilized Adaptive Loss and Residual-Based Collocation for Physics-Informed Neural Networks
2026-03-03 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors looked at improving a type of AI method called Physics-Informed Neural Networks (PINNs) that solve math problems related to physics, especially when those problems are tough or have sudden changes. They focused on making the training process more balanced and the solutions more accurate by introducing new techniques that adjust how the model learns and where it focuses more effort. Testing on specific equations showed their new method made solutions much more accurate, lowering errors significantly compared to the traditional way. They also confirmed their results by comparing them with trusted traditional methods.
Physics-Informed Neural Networkspartial differential equationsviscous Burgers' equationAllen-Cahn equationadaptive loss balancingcollocation schemestiffnessshock-dominated dynamicsrelative L2 errorfinite difference solver
Authors
Divyavardhan Singh, Shubham Kamble, Dimple Sonone, Kishor Upla
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.