A Multi-Objective Optimization Framework for Adaptive Weighting in Physics-Informed Machine Learning

Training physics-informed neural networks (PINNs) can be viewed as a multi-task optimization problem, where data-driven and physics-driven loss functions must be simultaneously minimized, despite the potential competition between them. Manually tuning the weight coefficients for various loss terms in PINNs is often time-consuming and lacks a systematic approach. To address this challenge, this work proposes an adaptive loss balancing framework for PINNs, using multi-objective optimization (MOO) algorithms to dynamically balance competing loss terms during training. Specifically, the Non-dominated Sorting Genetic Algorithm II (NSGA-II) is integrated into the PINN training process to explore the Pareto front of the multiple objectives. A novel variance-aware relative improvement (VARI) weighting method is proposed to translate Pareto-optimal information into adaptive loss weights. The proposed MOO-VARI method is validated through several examples, where the results show that the MOO-VARI PINN consistently outperforms standard PINN and other state-of-the-art adaptive weighting strategies in terms of convergence speed, predictive accuracy, and parameter estimation performance.