Learning collective variables that preserve transition rates

TL;DR

This paper develops a principled approach for designing collective variables using neural networks that accurately preserve transition rates in high-dimensional systems, with applications to molecular dynamics.

Contribution

It introduces a new method combining manifold learning and group invariance for neural network-based CV design, ensuring preservation of transition rates.

Findings

Successfully constructed CVs for butane with less than 10% error in transition rate

Provided empirical evidence questioning the need for positive definiteness in diffusion tensors

Highlighted the importance of light atoms in effective CV design

Abstract

Collective variables (CVs) play a crucial role in capturing rare events in high-dimensional systems, motivating the continual search for principled approaches to their design. In this work, we revisit the framework of quantitative coarse graining and identify the orthogonality condition from Legoll and Lelievre (2010) as a key criterion for constructing CVs that accurately preserve the statistical properties of the original process. We establish that satisfaction of the orthogonality condition enables error estimates for both relative entropy and pathwise distance to scale proportionally with the degree of scale separation. Building on this foundation, we introduce a general numerical method for designing neural network-based CVs that integrates tools from manifold learning with group-invariant featurization. To demonstrate the efficacy of our approach, we construct CVs for butane and achieve a CV that reproduces the anti-gauche transition rate with less than ten percent relative error. Additionally, we provide empirical evidence challenging the necessity of uniform positive definiteness in diffusion tensors for transition rate reproduction and highlight the critical role of light atoms in CV design for molecular dynamics.