Coarse-graining errors and numerical optimization using a relative entropy framework

Abstract

The ability to generate accurate coarse-grained models from reference fully atomic (or otherwise "first-principles") ones has become an important component in modeling the behavior of complex molecular systems with large length and time scales. We recently proposed a novel coarse-graining approach based upon variational minimization of a configuration-space functional called the relative entropy, S(rel), that measures the information lost upon coarse-graining. Here, we develop a broad theoretical framework for this methodology and numerical strategies for its use in practical coarse-graining settings. In particular, we show that the relative entropy offers tight control over the errors due to coarse-graining in arbitrary microscopic properties, and suggests a systematic approach to reducing them. We also describe fundamental connections between this optimization methodology and other coarse-graining strategies like inverse Monte Carlo, force matching, energy matching, and variational mean-field theory. We suggest several new numerical approaches to its minimization that provide new coarse-graining strategies. Finally, we demonstrate the application of these theoretical considerations and algorithms to a simple, instructive system and characterize convergence and errors within the relative entropy framework.

Enhanced, targeted sampling of high-dimensional free-energy landscapes using variationally enhanced sampling, with an application to chignolin

“…The functional Ω½V is convex with respect to V ðsÞ which means that this is the global minimum. Minimizing this functional is equivalent to minimizing the Kullback-Leibler divergence between the sampled distribution and the target distribution (25)(26)(27)(28). Eqs.…”

Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems

“…Furthermore even when it is available in the form of a Gibbs state it will require computations that will typically involve a full Hamiltonian, 7,22 .…”

“…Aside of this rigorous numerical analysis direction, entropy-based computational techniques were also developed and used for constructing approximations of coarse-grained (effective) potentials for models of large biomolecules and polymeric systems (fluids, melts). Optimal parametrization of effective potentials based on minimizing the relative entropy between equilibrium Gibbs states, e.g., 3,6,7 , extended previously developed inverse Monte Carlo methods, primarily based on force matching approaches, used in coarsegraining of macromolecules (see, e.g., 30,38 ). In 13 an extension to dynamics is proposed in the context of FokkerPlanck equations, by considering the corresponding relative entropy for discrete-time approximations of the transition probabilities.…”

From Atomistic to Systematic Coarse-Grained Models for Molecular Systems

“…The curves are very close to each other; however there are small differences, in particular in small distances, close to the first maximum. Note that theoretically it is expected that the RE outcome, at the level of g(R), should agree with the IBI one [15]. We should report here that we have calculated the CG potential derivatives appearing in the Jacobian and Hessian in the Newton-Raphson scheme by direct sampling during the corresponding CG run.…”

“…Methods such as inverse Monte Carlo (IMC), direct inverse Boltzmann (DBI) and iterative inverse Boltzmann (IBI) [11,12,1], force-matching [13,14], and relative entropy minimization [15] provide optimal parameterizations of approximate coarse-grained models by considering a pre-selected set of observables and by then minimizing a cost functional over the parameter space,…”