Bayesian, Frequentist, and Information Geometry Approaches to Parametric Uncertainty Quantification of Classical Empirical Interatomic Potentials

摘要

Uncertainty quantification (UQ) is an increasingly important part of materials modeling. In this paper, we consider the problem of quantifying parametric uncertainty in classical empirical interatomic potentials (IPs). Previous work based on local sensitivity analysis using the Fisher Information has shown that IPs are sloppy, i.e., are insensitive to coordinated changes of many parameter combinations. We confirm these results and further explore the non-local statistics in the context of sloppy model analysis using both Bayesian (MCMC) and Frequentist (profile likelihood) methods. We interface these tools with the Knowledgebase of Interatomic Models (OpenKIM) and study three models based on the Lennard-Jones, Morse, and Stillinger-Weber potentials, respectively. We confirm that IPs have global properties similar to those of sloppy models from fields such as systems biology, power systems, and critical phenomena. These models exhibit a low effective dimensionality in which many of the parameters are unidentifiable, i.e., do not encode any information when fit to data. Because the inverse problem in such models is ill-conditioned, unidentifiable parameters present challenges for traditional statistical methods. In the Bayesian approach, Monte Carlo samples can depend on the choice of prior in subtle ways. In particular, they often "evaporate" parameters into high-entropy, sub-optimal regions of the parameter space. For profile likelihoods, confidence regions are extremely sensitive to the choice of confidence level. To get a better picture of the relationship between data and parametric uncertainty, we sample the Bayesian posterior at several sampling temperatures and compare the results with those of Frequentist analyses. In analogy to statistical mechanics, we classify samples as either energy-dominated, i.e., characterized by identifiable parameters in constrained (ground state) regions of parameter space, or entropy-dominated, i.e., characterized by unidentifiable (evaporated) parameters. We complement these two pictures with information geometry to illuminate the underlying cause of this phenomenon. In this approach, a parameterized model is interpreted as a manifold embedded in the space of possible data with parameters as coordinates. We calculate geodesics on the model manifold and find that IPs, like other sloppy models, have 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 UQ analysis and further suggest simplified, less-sloppy models.