On finding optimal collective variables for complex systems by minimizing the deviation between effective and full dynamics

TL;DR

This paper develops a theoretical framework for selecting optimal collective variables in complex systems by minimizing the deviation between effective and full dynamics, providing criteria and error estimates for better system understanding.

Contribution

It introduces strict criteria for optimal collective variable selection based on effective dynamics properties and links data-driven methods to these theoretical insights.

Findings

Transition density of optimal variables solves a relative entropy minimization problem.

Many data-driven approaches learn quantities related to effective dynamics.

Error estimates show how optimal variables improve approximation of system timescales.

Abstract

This paper is concerned with collective variables, or reaction coordinates, that map a discrete-in-time Markov process $X_n$ in $\mathbb{R}^d$ to a (much) smaller dimension $k \ll d$. We define the effective dynamics under a given collective variable map $\xi$ as the best Markovian representation of $X_n$ under $\xi$. The novelty of the paper is that it gives strict criteria for selecting optimal collective variables via the properties of the effective dynamics. In particular, we show that the transition density of the effective dynamics of the optimal collective variable solves a relative entropy minimization problem from certain family of densities to the transition density of $X_n$. We also show that many transfer operator-based data-driven numerical approaches essentially learn quantities of the effective dynamics. Furthermore, we obtain various error estimates for the effective dynamics in approximating dominant timescales / eigenvalues and transition rates of the original process $X_n$ and how optimal collective variables minimize these errors. Our results contribute to the development of theoretical tools for the understanding of complex dynamical systems, e.g. molecular kinetics, on large timescales. These results shed light on the relations among existing data-driven numerical approaches for identifying good collective variables, and they also motivate the development of new methods.