Generic bounds on the approximation error for physics-informed (and) operator learning

TL;DR

This paper introduces a comprehensive framework for deriving rigorous bounds on the approximation errors of physics-informed neural networks and operator learning architectures, ensuring efficient PDE solutions.

Contribution

It provides the first rigorous bounds on physics-informed operator learning and demonstrates that PINNs and related methods mitigate the curse of dimensionality in nonlinear PDE approximation.

Findings

Derived the first rigorous bounds for physics-informed operator learning.

Showed PINNs and related architectures mitigate the curse of dimensionality.

Provided a general framework for approximation error analysis in PDE learning.

Abstract

We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more involved learning architectures for PDEs. We illustrate the general framework by deriving the first rigorous bounds on the approximation error of physics-informed operator learning and by showing that PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of dimensionality in approximating nonlinear parabolic PDEs.