Relative Entropy Methods for Calculating Committors

TL;DR

This work addresses the challenge of computing reactive trajectories and committor functions under overdamped Langevin dynamics, focusing on rare transitions between metastable sets. It introduces a relative-entropy–based loss that measures how well an approximate committor induces the transition-path law, and develops a computable change-of-measure formula between the exact and approximate transition-path processes that remarkably does not require knowledge of the exact committor. The authors propose a stochastic-gradient-descent training approach to minimize the KL divergence between the exact and approximate path measures, along with model-selection criteria based on entropy differences, and they provide practical numerical strategies for representing committors and handling singular boundary behavior. Together, these contributions enable efficient assessment, training, and selection of committor approximations, with potential impact on enhanced sampling and accurate estimation of transition statistics in molecular systems.

Abstract

Motivated by challenges arising in molecular simulation, we analyze and develop methods of computing reactive trajectories and committor functions for systems described by the overdamped Langevin dynamics. Our main technical advance is a new loss function that measures the accuracy of approximations to the committor function related to a given chemical reaction or other rare transition event. Our loss admits a simple interpretation in terms of the distribution of reactive trajectories, and it can be computed in practice to compare the accuracies of different approximations of the committor. We also derive a method of calculating committors by direct minimization of the loss via stochastic gradient descent.

Figures (2)

• Figure 1: Schematic illustration of reactive trajectories, entrance times, and exit times. The black line depicts a trajectory of overdamped Langevin dynamics. The bold red segments depict reactive trajectories. Here, $\tau_{B,0}$ is the first hitting time of $B$, $\tau_{A,1}$ is the first hitting time of $A$ after $\tau_{B,0}$, $\tau_{B,1}$ is the first hitting time of $B$ after $\tau_{A,1}$, and so forth. The $k$'th exit time $\sigma_{A,k}$ is the last time before $\tau_{B,k}$ that the process was in $A$.
• Figure 2: A trajectory of the overdamped Langevin dynamics for a simple two-dimensional model potential. The blue curves are contours of the potential. The trajectory depicted here was initialized from the Boltzmann distribution, i.e. in equilibrium. Its starting point happened to be in the set $A$ on the left side of the figure. The trajectory was terminated on hitting the boundary of $B$ for the first time. The bold red curve depicts the end of the trajectory from the last time it left $A$ to the first time it entered $B$; i.e. the reactive part of the trajectory. The faint red curve depicts the beginning of the trajectory up to the last time it left $A$.

Theorems & Definitions (30)

• Remark 1
• Lemma 1
• proof
• Remark 2
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 3: Weak Solutions of the Approximate Transition Path Equation
• Lemma 4