Incorporating Continuous Dependence Qualifies Physics-Informed Neural Networks for Operator Learning

TL;DR

This paper introduces cd-PINN, an extension of physics-informed neural networks that incorporates continuous dependence of PDE solutions, significantly improving their generalization and accuracy in operator learning tasks with limited data.

Contribution

The paper proposes a novel extension of PINNs by integrating continuous dependence information, enhancing their capability for operator learning in PDEs.

Findings

cd-PINN achieves 1-3 orders of magnitude lower test MSE than DeepONet and FNO.

Incorporating continuous dependence improves PINNs' generalization in high-dimensional PDEs.

The method provides a simple way to qualify PINNs for operator learning.

Abstract

Physics-informed neural networks (PINNs) have been proven as a promising way for solving various partial differential equations, especially high-dimensional ones and those with irregular boundaries. However, their capabilities in real applications are highly restricted by their poor generalization performance. Inspired by the rigorous mathematical statements on the well-posedness of PDEs, we develop a novel extension of PINNs by incorporating the additional information on the continuous dependence of PDE solutions with respect to parameters and initial/boundary values (abbreviated as cd-PINN). Extensive numerical experiments demonstrate that, with limited labeled data, cd-PINN achieves 1-3 orders of magnitude lower in test MSE than DeepONet and FNO. Therefore, incorporating the continuous dependence of PDE solutions provides a simple way for qualifying PINNs for operator learning.