Accuracy and Robustness of Weight-Balancing Methods for Training PINNs

Abstract: Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for integrating physics-based models with data by minimizing both data and physics losses. However, this multi-objective optimization problem is notoriously challenging, with some benchmark problems leading to unfeasible solutions. To address these issues, various strategies have been proposed, including adaptive weight adjustments in the loss function. In this work, we introduce clear definitions of accuracy and robustness in the context of PINNs and propose a novel training algorithm based on the Primal-Dual (PD) optimization framework. Our approach enhances the robustness of PINNs while maintaining comparable performance to existing weight-balancing methods. Numerical experiments demonstrate that the PD method consistently achieves reliable solutions across all investigated cases, even in the low-data regime, and can be easily implemented, facilitating its practical adoption. The code is available at https://github.com/haoming-SHEN/Accuracy-and-Robustness-of-Weight-Balancing-Methods-for-Training-PINNs.git.