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

Abstract

In this paper we focus on the development of new methods suitable for efficient and reliable coarse-graining of non-equilibrium molecular systems. In this context, we propose error estimation and controlled-fidelity model reduction methods based on Path-Space Information Theory, combined with statistical parametric estimation of rates for non-equilibrium stationary processes. The approach we propose extends the applicability of existing information-based methods for deriving parametrized coarse-grained models to Non-Equilibrium systems with Stationary States (NESS). In the context of coarse-graining it allows for constructing optimal parametrized Markovian coarse-grained dynamics within a parametric family, by minimizing information loss (due to coarse-graining) on the path space. Furthermore, we propose an asymptotically equivalent methodrelated to maximum likelihood estimators for stochastic processes-where the coarse-graining is obtained by optimizing the information content in path space of the coarse variables, with respect to the projected computational data from a fine-scale simulation. Finally, the associated path-space Fisher Information Matrix can provide confidence intervals for the corresponding parameter estimators. We demonstrate the proposed coarse-graining method in (a) non-equilibrium systems with diffusing interacting particles, driven by out-ofequilibrium boundary conditions, as well as (b) multi-scale diffusions and their well-studied corresponding stochastic averaging limits, comparing them to our proposed methodologies.

Data-driven, variational model reduction of high-dimensional reaction networks

“…The purpose of this section is to review main information-based tools used in the variational model reduction approach for stochastic systems introduced in [36,44], and to present the mathematical framework of this work.…”

“…We start this example by noticing that 10 out of 87 parameters accumulate at least 95% (precisely, 96.524%) of the total information as per the pFIM diagonal (44), as shown in Figure 9. Total information of a set of parameters is a natural and intuitive measure that allows to choose a meaningful set of parameters for information-based model reduction.…”

“…In this work we develop a method that, given a reaction network with path distribution P 0:T , i.e., a probability over the set of all possible dynamics over a time interval, finds an element from a parameterized family of distributions that is close to P 0:T . The novelty of this work is two-fold, (a) it extends the coarse-graining path-space information theory methods of [44,36] to achieve model reductions of complex reaction networks, and (b) obtains the most efficient model reduction in (a) by applying a path-space sensitivity analysis technique presented in [71,72], thus identifying the most sensitive model parameters. More specifically, in (a) we consider the path probability distribution P 0:T of a high-dimensional reaction network, i.e., a probability over the set of all possible dynamics (time series) on a time interval [0, T ]; then we seek an element from a parameterized family of reduced models distributions that is closest to P 0:T with respect to a loss function, which in this case is the relative entropy: min θ∈Θ R(P 0:T Q θ 0:T ) .…”

From Atomistic to Systematic Coarse-Grained Models for Molecular Systems

“…Methods such as inverse Monte Carlo [1], inverse Boltzmann [2], force matching [3], relative entropy [4], provide parameterizations of coarse-grained effective potentials at equilibrium by minimizing a fitting functional over a parameter space. Then, we further extend these studies using path-space methods (relative entropy rate) for coarse-graining and uncertainty quantification for non-equilibrium processes, [5,6,7,8].…”

Kullback--Leibler Approximation for Probability Measures on Infinite Dimensional Spaces

“…Recently the methodology has been used for the coarse-graining of stochastic lattice systems [22], simple models for data assimilation [2,3], the study of models in ocean-atmosphere science [25,17], and molecular dynamics [21]. However, none of this applied work has studied the underlying calculus of variations problem, which is the basis for the algorithms employed.…”