Transferable Physics-Informed Representations via Closed-Form Head Adaptation

2026-04-23

AI Summary

This work addresses the limited generalization and computational inefficiency of conventional physics-informed neural networks (PINNs) when solving new partial differential equation (PDE) problems under extreme data scarcity. To overcome these challenges, the authors propose Pi-PINN, a novel framework that integrates a closed-form pseudoinverse head adaptation mechanism with multi-task learning and physics-constrained loss functions to construct transferable physical representations. By operating within a shared embedding space, Pi-PINN enables rapid and accurate solutions for both known and previously unseen PDEs without requiring additional training data—achieving high-fidelity predictions from as few as two samples. Empirical evaluations on Poisson, Helmholtz, and Burgers equations demonstrate that Pi-PINN accelerates convergence by 100–1000 times and reduces relative errors by 10–100 times compared to standard PINNs.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Search and Optimization: Learning to Search
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• physics-informed neural networks
• partial differential equations
• generalization
• transfer learning
• unseen PDE instances

Innovation

• Physics-Informed Neural Networks
• Transferable Representations
• Closed-Form Adaptation
• Pseudoinverse Learning
• Partial Differential Equations