Optimal Reaction Coordinates: Variational Characterization and Sparse Computation

TL;DR

This paper introduces a variational framework for identifying optimal reaction coordinates in high-dimensional stochastic systems, enabling efficient, data-sparse computation that accurately captures long-term dynamics.

Contribution

It provides a novel variational characterization of optimal reaction coordinates and establishes conditions for their existence, facilitating efficient computation using machine learning.

Findings

Reproduces known insights for slow-fast and metastable systems.

Loss function evaluation scales with low-dimensional complexity.

Framework guarantees good approximation of long-term dynamics.

Abstract

Reaction Coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function and reduced models based on them guarantee very good approximation of the long-term dynamics of the original high-dimensional process. We show that, for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, the numerical effort required to evaluate the loss function scales only with the complexity of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques.