Estimation of error in observables of coarse-grained models of atomic systems

Abstract

Background: The use of coarse-grained approximations of atomic systems is the most common methods of constructing reduced-order models in computational science. However, the issue of central importance in developing these models is the accuracy with which they approximate key features of the atomistic system. Many methods have been proposed to calibrate coarse-grained models so that they qualitatively mimic the atomic systems, but these are often based on heuristic arguments. Methods: A general framework for deriving a posteriori estimates of modeling error in coarse-grained models of key observables in atomistic systems is presented. Such estimates provide a new tool for model validation analysis. The connection of error estimates with relative information entropy of observables and model predictions is explained for so-called misspecified models. The relationship between model plausibilities and Kullback-Leibler divergence between the true parameters and model predictions is summed up in several theorems. Results: Numerical examples are presented in this paper involving a family of coarsegrained models of a polyethylene chain of united atom monomers. Numerical results suggest that the proposed methods of error estimation can be very good indications of the error inherent in coarse-grained models of observables in the atomistic systems. Also, new theorems relating the Kullback-Leibler divergence between model predictions and observations to measures of model plausibility are presented. Conclusions: A formal structure for estimating errors produced by coarse-graining atomistic models is presented. Numerical examples confirm that the estimates are in agreement with exact errors for a simple class of materials. Errors measured in the D KL -divergence can be related to computable model plausibilities. The results should provide a powerful framework for assessing the validity and accuracy of coarse-grained models.

A Predictive Discrete-Continuum Multiscale Model of Plasticity With Quantified Uncertainty

“…Such momentum is because these methods offer general frameworks for predictive modeling while providing means to portray uncertainty. Here, we summarize our Bayesian calibration process, as described in [93] and implemented in [98,99,39,94,32] for predictive modeling of various physical systems. Consider θ to be a vector of model parameters and D to be the observational (training) data.…”

Multilevel Bayesian Parameter Estimation in the Presence of Model Inadequacy and Data Uncertainty

“…Therefore, the marginalization in (40) is strictly performed on a subset of Ω R * that satisfy (28). By contrast, not all possible realizations, R * ∈ Ω R * , of R * in the hierarchical likelihood function of (60) have to necessarily satisfy (28). For the construction of this likelihood function, we have already assumed that the physical model is inadequate in describing the truth, R. Hence, the marginalization spans the entire sampling space of R * , which is Ω R * .…”

“…Once the parameters of a physical model are constrained, the proposed physical model has to be verified and its predictions validated against a new independent dataset. Extensive literature already exists on the topic of model verification and validation [e.g., 3,4,7,8,24,44,51,56,60,66] as well as on decision theory [for elegant reviews from a Bayesian perspective, see 39,40,47]. The validated model can be then used to make predictions of the Quantities of Interest (QoI), the precise physical features of the response of the system targeted in the simulation.…”