Continual-Learning Physics-Informed Neural Networks for Parameterized Partial Differential Equations

Physics-informed neural networks (PINNs) are powerful tools for solving partial differential equations (PDEs) without large observational datasets. However, when extended to parameterized PDEs (ParamPINNs)—where a single model must represent a family of solutions across a parameter domain—they often suffer from inefficient training, imbalanced accuracy across parameters, and overfitting to a limited set of training tasks. This paper asks how to train these models more effectively under limited computational resources and without relying on external observational data.

The authors introduce CL-PINN, a continual-learning approach that treats different parameter values as sequential tasks. The framework includes three primary components:

1. Active Parameter Selection: Uses Bayesian optimization (BO) to iteratively select the most informative parameter tasks, reducing the need for exhaustive grid searches.
2. Dynamic Loss Weighting: Adjusts task-wise weights during training to mitigate multiscale solution behavior and ensure balanced convergence across different physical parameters.
3. Sparse Physics-Constrained Replay: Employs experience replay to retain knowledge of earlier tasks, preventing catastrophic forgetting as the model is updated with new parameter instances.

Evaluations across five benchmarks—including one continuous function and four parameterized PDEs—demonstrate that CL-PINN significantly improves training efficiency compared to grid-greedy baselines. The Bayesian selection process reduces the number of objective-loss queries required to cover the parameter space, while the sparse replay mechanism successfully maintains accuracy on earlier tasks. Overall, the model achieves higher and more balanced solution accuracy across the parameter domain, providing a scalable path for creating reusable physics-informed surrogates in engineering applications.