Koopman operators improve rare state transition sampling in molecular simulations
Transition Path Sampling Using Koopman Operators and Exit-Time Optimal Control
Machine Learning
Summary
Transitions between stable states in molecules or complex systems can be very rare and hard to study because they happen across high energy barriers. The authors developed a new method using Koopman operators, a mathematical tool that helps identify stable regions and transition probabilities without needing prior transition data. They reformulated the transition sampling task as an optimal control problem with an adjustable exit time, leading to a controller that efficiently guides simulations toward target states. Their approach was tested on model systems and a small molecule, greatly increasing the chance of observing transitions within practical time frames.
What this means in practice
- •For computational chemists: Enhance molecular simulations to efficiently sample rare transitions between conformations without extensive biased trial runs.
- •For chemical process engineers: Improve modeling of state changes in complex chemical systems to better predict reaction pathways and optimize process conditions.
Authors
Boya Hou, Shane Wang, Siddharth Ambekar, Maxim Raginsky, Olgica Milenkovic
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.