Transition Path Sampling Using Koopman Operators and Exit-Time Optimal Control

2026-10-07

AI Summary

This study addresses the challenge of sampling metastable transitions under high free-energy barriers and the lack of performance guarantees in existing machine learning approaches. We propose an optimal control-enhanced sampling framework based on the Koopman operator. This method identifies metastable states without requiring trajectory information and exploits the linearity of the Koopman operator to estimate committor functions. Sampling is formulated as an exit-time optimal control problem, where reproducing kernel Hilbert space (RKHS) approximation reduces the complex optimization to a single quadratic program for efficiently solving the closed-loop controller. Evaluated on double-well and alanine dipeptide models, the proposed approach achieves target arrival rates of 99.8% and 93%, respectively, significantly outperforming baseline methods while offering both computational efficiency and theoretical guarantees.

Abstract

Sampling transitions between metastable states is a central problem in dynamical systems theory and molecular dynamics in particular. A key challenge is the existence of high free-energy barriers that separate the states, making transitions extremely rare. Recent machine learning-based methods cast transition path sampling (TPS) as an optimal stochastic control (OSC) problem over a fixed time horizon, and parameterize the drift bias via a neural network trained by simulation-in-the-loop, requiring repeated biased rollouts. To address computational and performance guarantee issues of these models, we propose a new approach for the problem based on Koopman operators. Because Koopman operators are linear, their leading eigenfunctions reveal the metastable sets and provide an estimate of the committor function with no transition path information required. Furthermore, we formulate TPS as an OSC problem up to an exit time. Our time horizon is the first hitting time of the target set, and our running cost penalizes time spent in nonreactive regions by encoding the estimated committor function. We derive the optimal controller in closed form and approximate it in a reproducing kernel Hilbert space (RKHS). This reduces the problem of constructing the optimal controller to solving a single equality-constrained quadratic program, whose solution can be characterized by a linear Karush-Kuhn-Tucker (KKT) system. On the two-channel double well and alanine dipeptide, our controller increases the fraction of trajectories reaching the target from 0% to 99.8% within 1000 steps, and from 0% to 93% within 1ps, respectively.

Problem

• Transition Path Sampling
• Metastable States
• Optimal Stochastic Control
• Free-Energy Barriers
• Molecular Dynamics

Innovation

• Transition Path Sampling
• Koopman Operators
• Optimal Stochastic Control
• Committor Function
• Reproducing Kernel Hilbert Space