Abstract
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. 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. Numerical examples are presented in this paper involving a family of coarse-grained 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. 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.
Highlights
The use of coarse-grained approximations of atomic systems is the most common methods of constructing reduced-order models in computational science
Coarse-grained-reduced order models The most common method of constructing reduced-order models in all of computational science involves the use of coarse-grained models of atomic systems, whereby systems of atoms are aggregated into “beads”, or “super atoms”, or molecules to reduce the number of degrees of freedom and to lengthen the time scales in which the evolution of events are simulated
The use of coarse-grained (CG) approximations has been prevalent in molecular dynamics (MD) simulations for many decades Comprehensive reviews of a large segment of the literature on CG models was recently given by Noid [1] and Li et al [2], and an application to semiconductor nano-manufacturing is discussed in Farrell et al [3]
Summary
The use of coarse-grained approximations of atomic systems is the most common methods of constructing reduced-order models in computational science. 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. We focus on standard molecular dynamics models of micro-canonical ensemble (NVE) thermodynamics, and we call upon the theory of model adaptivity and error estimation laid down in [4] and [5]. In this particular setting, new estimates are obtained when the information entropy of Shannon [6] is used as a quantity of interest. This leads to methods for estimating CG-model parameters that involve the Kullback-Leibler divergence between probability densities of observables in the AA and CG systems
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