Transferable Physics-Informed Representations via Closed-Form Head Adaptation

TL;DR

This paper introduces Pi-PINN, a transfer learning framework for physics-informed neural networks that enables rapid, accurate solutions to unseen PDEs using closed-form head adaptation, significantly improving efficiency and generalization.

Contribution

The paper proposes a novel transferable PINN framework with closed-form head adaptation, allowing fast solving of new PDEs without additional training data.

Findings

Pi-PINN achieves 100-1000x faster predictions than typical PINNs.

Pi-PINN attains 10-100x lower relative error compared to data-driven models.

Pi-PINN generalizes across multiple PDEs like Poisson, Helmholtz, and Burgers' equations.

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.