Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equations

Physics-informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi-PINN). Pi-PINN learns a transferable physics-informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed-form head adaptation using a least-squares-optimal pseudoinverse under PDE constraints. We further investigate the synergies between data-driven multi-task learning loss and physics-informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi-PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics-informed solutions without requiring any data for unseen instances. Pi-PINN can produce predictions 100-1000 times faster than a typical PINN, while producing predictions with 10-100 times lower relative error than a typical data-driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed-form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.

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.

Physics-informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by embedding physical laws into the loss function. In parameterized PDE families, variations in coefficients or boundary/initial conditions define distinct tasks. This makes training individual PINNs for each task computationally prohibitive, while cross-task transfer can be sensitive to task heterogeneity. While meta-learning can reduce retraining cost, existing methods often rely on a single global initialization and may suffer from negative transfer, particularly under feature-scarce coordinate inputs and limited training-task availability. We propose the Learning-Affinity Adaptive Modular Physics-Informed Neural Network (LAM-PINN), a compositional framework that leverages task-specific learning dynamics. LAM-PINN combines PDE parameters with learning-affinity metrics from brief transfer sessions to construct a task representation and cluster tasks even with coordinate-only inputs. It decomposes the model into cluster-specialized subnetworks and a shared meta network, and learns routing weights to selectively reuse modules instead of relying on a single global initialization. Across three PDE benchmarks, LAM-PINN achieves an average 19.7-fold reduction in mean squared error (MSE) on unseen tasks using only 10% of the training iterations required by conventional PINNs. These results indicate its effectiveness for generalization to unseen configurations within bounded design spaces of parameterized PDE families in resource-constrained engineering settings.