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

2026-08-05

AI Summary

This work addresses key limitations of conventional parametric physics-informed neural networks—namely, suboptimal training efficiency, imbalanced accuracy across the parameter domain, and poor generalization to unsampled parameters. The authors reformulate the solution of multiparametric partial differential equations as a continual learning task and introduce a novel framework that integrates Bayesian optimization-based active sampling, task-level dynamic loss weighting, sparse physics-constraint replay, and optional parameter-specific subnetworks. Operating without observational data, the method efficiently leverages limited computational resources, substantially reducing the number of target loss queries while effectively mitigating catastrophic forgetting. Empirical evaluations across five benchmark problems demonstrate consistently higher and more uniformly distributed solution accuracy compared to existing approaches.

Abstract

Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a family of PDE solutions over a parameter domain. Existing ParamPINNs, however, still face inefficient training, uneven accuracy across parameters, and overfitting to a limited set of sampled parameter tasks, which can impair generalization to unsampled parameters. To address these issues, we propose a continual-learning physics-informed neural network (CL-PINN), which treats PDE instances at different parameter values as related tasks and learns them sequentially. CL-PINN combines Bayesian-optimization-based active parameter selection, task-wise dynamic loss weighting, sparse physics-constrained replay, and an optional parameter subnetwork to improve task allocation and knowledge retention under bounded active-task capacity. It requires no observational data and is designed to solve parameterized PDEs over relatively broad parameter domains under limited computational resources. Multi-seed evaluations on five benchmarks, including one continuous function and four parameterized PDEs, show that Bayesian selection substantially reduces objective-loss queries relative to grid-greedy search, while sparse replay mitigates forgetting of earlier tasks. Under the prescribed within-case resource protocols, CL-PINN generally provides higher and more balanced solution accuracy than fixed-sampling and grid-greedy baselines. CL-PINN offers a practical route toward learning PDE solutions that generalize across physical parameters and has the potential to support reusable physics-informed surrogates for large-scale engineering parameter studies.

Problem

• parameterized PDEs
• physics-informed neural networks
• continual learning
• generalization
• overfitting

Innovation

• continual learning
• physics-informed neural networks
• parameterized PDEs
• Bayesian optimization
• sparse replay