Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI

TL;DR

This paper addresses the inherent uncertainties in physics-informed inverse problems, proposing a framework to quantify and analyze these uncertainties, and demonstrating how geometric constraints can ensure unique identification of physical models.

Contribution

It introduces a novel framework for quantifying uncertainties in PIML coefficient estimation and shows how geometric constraints can lead to unique model identification.

Findings

Uncertainties exist in coefficient function estimation in PIML.

Geometric constraints can reduce uncertainties and enable unique model identification.

The framework successfully applied to a magnetohydrodynamics model.

Abstract

Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data-driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data-driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.