Towards a Foundation Model for Physics-Informed Neural Networks: Multi-PDE Learning with Active Sampling

TL;DR

This paper proposes a unified foundation model for Physics-Informed Neural Networks capable of solving multiple PDEs, enhanced with active learning to improve sample efficiency and accuracy across diverse physical systems.

Contribution

It introduces a multi-PDE PINN framework trained on various equations, combined with active learning for efficient sampling, advancing generalizability and reducing computational costs.

Findings

Active learning improves solution accuracy with fewer samples.

The multi-PDE PINN effectively learns diverse physical dynamics.

Targeted sampling enhances training efficiency across multiple PDEs.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training. However, traditional PINN models are typically designed for single PDEs, limiting their generalizability across different physical systems. In this work, we explore the potential of a foundation PINN model capable of solving multiple PDEs within a unified architecture. We investigate the efficacy of a single PINN framework trained on four distinct PDEs-the Simple Harmonic Oscillator (SHO), the 1D Heat Equation, the 1D Wave Equation, and the 2D Laplace Equation, demonstrating its ability to learn diverse physical dynamics. To enhance sample efficiency, we incorporate Active Learning (AL) using Monte Carlo (MC) Dropout-based uncertainty estimation, selecting the most informative training samples iteratively. We evaluate different active learning strategies, comparing models trained on 10%, 20%, 30%, 40%, and 50% of the full dataset, and analyze their impact on solution accuracy. Our results indicate that targeted uncertainty sampling significantly improves performance with fewer training samples, leading to efficient learning across multiple PDEs. This work highlights the feasibility of a generalizable PINN-based foundation model, capable of adapting to different physics-based problems without redesigning network architectures. Our findings suggest that multi-PDE PINNs with active learning can serve as an effective approach for reducing computational costs while maintaining high accuracy in physics-based deep learning applications.