Continual-Learning Physics-Informed Neural Networks for Parameterized Partial Differential Equations
Xujia Chen, Xinyue Hu, Letian Chen, Yi Liu, Wenhui Fan
cs.LG
Submitted: 2026-08-05
Comments: 124 pages in total, including the main text (63 pages, 25 figures, and 16 tables) and supplementary material (61 pages, 30 figures, and 17 tables)
Code: https://github.com/pigofmomo/CLPINN
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
The gist: Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets.
Terminology
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.
Sources
- DeepCFD: Efficient Steady-State Laminar Flow Approximation with Deep Convolutional Neural Networks
- GrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equations
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
- Fourier Neural Operator for Parametric Partial Differential Equations
- Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications
- GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs
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