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

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 optimal stochastic control (OSC) over a fixed time horizon, and parameterize the drift bias with a neural network trained by simulation-in-the-loop, where training requires repeated biased rollouts. We instead develop a new operator-theoretic approach for the problem based on the Koopman operator. 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. 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 convex quadratic program, whose solution is characterized by a linear 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 $1,000$ steps, and from $0\%$ to $93\%$ within $1\mathrm{ps}$, respectively.