Bayesian, frequentist, and information geometric approaches to parametric uncertainty quantification of classical empirical interatomic potentials

TL;DR

This paper compares Bayesian and frequentist methods for quantifying parametric uncertainty in classical interatomic potentials, revealing their inherent sloppiness and proposing geometric insights for improved modeling.

Contribution

It introduces an information geometric framework to analyze and address the sloppiness and unidentifiability in interatomic potential models, enhancing uncertainty quantification.

Findings

Interatomic potentials are typically sloppy and unidentifiable.

Information geometry reveals the global structure of these models.

New parameterizations can improve model stability and interpretability.

Abstract

In this paper, we consider the problem of quantifying parametric uncertainty in classical empirical interatomic potentials (IPs) using both Bayesian (Markov Chain Monte Carlo) and frequentist (profile likelihood) methods. We interface these tools with the Open Knowledgebase of Interatomic Models and study three models based on the Lennard-Jones, Morse, and Stillinger--Weber potentials. We confirm that IPs are typically sloppy, i.e., insensitive to coordinated changes in some parameter combinations. Because the inverse problem in such models is ill-conditioned, parameters are unidentifiable. This presents challenges for traditional statistical methods, as we demonstrate and interpret within both Bayesian and frequentist frameworks. We use information geometry to illuminate the underlying cause of this phenomenon and show that IPs have global properties similar to those of sloppy models from fields such as systems biology, power systems, and critical phenomena. IPs correspond to bounded manifolds with a hierarchy of widths, leading to low effective dimensionality in the model. We show how information geometry can motivate new, natural parameterizations that improve the stability and interpretation of uncertainty quantification analysis and further suggest simplified, less-sloppy models.