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

Boya Hou; Shane Wang; Siddharth Ambekar; Maxim Raginsky; Olgica Milenkovic

eess.SYOct 7, 2026

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

Authors: Boya Hou, Shane Wang, Siddharth Ambekar, Maxim Raginsky, Olgica Milenkovic

Organizations: University of Illinois Urbana-Champaign

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.

Figures & tables

Figure 1 : Two-channel double well. The potential has two minima at $(\pm\sqrt{5}/2,\,0)$ connected by two channels around $(0,\pm 1)$ . Following Algorithm 1 , we compute the Koopman eigenfunctions and extract the metastable cores $A_{\mathrm{left}}$ and $A_{\mathrm{right}}$ as described in Section 4.1 . Their minima are marked by the green and red stars, and the dashed red curve marks $\partial A_{\mathrm{right}}$ . We sample $1000$ uncontrolled and $1000$ controlled paths, all started at the minimum of $A_{\mathrm{left}}$ . (Left) Potential energy landscape. (Middle) Uncontrolled paths: THP( $T_{\mathrm{max}}$ ) = $0\%$ . (Right) Proposed method: THP( $T_{\mathrm{max}}$ ) = $99.8\%$ .

Panel of Figure 2 — Figure 2 : Müller-Brown potential, with a deep well at $(-0.558,1.442)$ (green star), a medium well at $(0.623,0.028)$ (red circle), and a shallow well at $(-0.050,0.467)$ . We aim to sample transitions from the deep well to the medium well within $T_{\mathrm{max}}=10\mathrm{ps}$ . (a) Uncontrolled paths: THP( $T_{\mathrm{max}}$ ) = $1.8\%$ . (b) TPS-DPS ( Seong et al., 2025 ) : THP( $T_{\mathrm{max}}$ ) = $100\%$ , but the paths run over the high energy barrier. (c) Proposed method: THP( $T_{\mathrm{max}}$ ) = $99.0\%$ using $\beta=1$ , with paths following the low-energy channel through the shallow well rather than crossing the barrier.

Panel of Figure 3 — Figure 3 : Alanine dipeptide in vacuum, shown on the free-energy surface over the backbone dihedral angles $(\phi,\psi)$ . We aim to sample the transition from $C5$ conformation (upper left) to the $C7_{\mathrm{ax}}$ conformation (red star) within $T_{\mathrm{max}}=1\mathrm{ps}$ . (a) Uncontrolled paths: $(T_{\mathrm{max}})=0\%$ (b) TPS-DPS: THP $(T_{\mathrm{max}})=100\%$ , obtained by training a neural-network bias with simulation in the loop ( $1000$ rollouts, ${\sim}10^{6}$ gradient updates). (c) Proposed method: THP $(T_{\mathrm{max}})=93\%$ , with a controller obtained in closed form. The setup and computational cost are reported in Appendix G.4 .

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Method	THP ( $\uparrow$ )	ETS ( $\downarrow$ )
	%	kJmol -1
Uncontrolled (1200K)	$0.0$	–
Uncontrolled (2400K)	$2.7$	$1.73\pm 0.42$
Uncontrolled (3600K)	$18.8$	$2.07\pm 0.46$
Uncontrolled (4800K)	$42.1$	$2.49\pm 0.66$
TPS-DPS (1200K)	$99.9$	$1.38\pm 0.16$

Appendix

Table 1 : Target hit percentage (THP) and energy of the transition state (ETS) for the 2-channel double well. Uncontrolled and controlled rows are measured in our setup; TPS-DPS values are as reported by Seong et al. (2025) .

Method	THP ( $\uparrow$ )	ETS ( $\downarrow$ )
	%	kJmol -1
Uncontrolled (150K)	$0.1$	$12.4$
Uncontrolled (200K)	$1.8$	$14.0\pm 1.3$
TPS-DPS (150K)	$100$	$15.9\pm 3.5$
TPS-DPS (200K)	$100$	$15.7\pm 1.9$
Proposed method (150K, $\beta=1$ )	$95.1$	$14.1\pm 2.1$

Appendix

Table 2 : Müller-Brown benchmark at

T=150\,\mathrm{K}

and

200\,\mathrm{K}

Panel of Figure 4 — Figure 4 : Alanine dipeptide on the Ramachandran plain of backbone dihedral angles $(\phi,\psi)$ . (Left) The first nontrivial Koopman eigenfunction $\psi_{1}$ which separates the $C5$ basin from $C7_{\mathrm{ax}}$ basin. (Right) The committor $q$ from $C5$ to $C7_{\mathrm{ax}}$ .

Panel of Figure 5 — Figure 5 : Sampled transition paths for alanine dipeptide in vacuum, same as Figure 3 in Section 5 . Paths start in the $C5$ basin (upper left) and end in the $C7_{\mathrm{ax}}$ basin (red star). (a) Uncontrolled paths. (b) TPS-DPS ( Seong et al., 2025 ) : THP $=100\%$ , obtained by training a neural-network bias with simulation in the loop ( $1000$ rollouts, ${\sim}10^{6}$ gradient updates). (c) Proposed method: THP $=93\%$ , with a closed-form controller.

Proposed method
Stage	Time (s)	Total (s)
Sampling	Direct evaluation of ( 3 )	180
Koopman eigenpairs	73.8
Committor $\hat{q}$	56.2
Controller $\hat{u}$	51.0
Training-based baselines

Appendix

Table 3 : Cost of obtaining the controller on alanine dipeptide on a single NVIDIA A100 GPU, excluding the subsequent path generation, which is common to all methods.

Panel of Figure 6 — Figure 6 : Four-well dynamics at $T=200\,\mathrm{K}$ . We aim to sample transitions from the source core at $(-1,-1)$ (green star) to the target core at $(1,1)$ (red star). (a) Uncontrolled paths: THP( $T_{\mathrm{max}}$ ) = $0.9\%$ . (b) TPS-DPS ( Seong et al., 2025 ) : THP( $T_{\mathrm{max}}$ ) = $81.6\%$ . (c) Proposed method with $\beta=2$ : THP( $T_{\mathrm{max}}$ ) = $94.7\%$ .

Method	THP ( $\uparrow$ )	ETS ( $\downarrow$ )
	%	kJmol -1
Uncontrolled (200K)	$0.9$	$10.6\pm 3.2$
Uncontrolled (300K)	$8.8$	$12.8\pm 3.6$
TPS-DPS (200K)	$81.6$	$8.6\pm.9$
TPS-DPS (300K)	$87.1$	$10.2pm.6$
Proposed method (200K, $\beta=1$ )	$73.2$	$9.5\pm 1.8$

Appendix

Table 4 : Four-well benchmark at

T=200\,\mathrm{K}

and

300\,\mathrm{K}

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Studying the dynamical behavior of a system often depends on characterizing how it transitions between long-lived states. Because such transitions are rare, observing them usually requires specialized enhanced sampling techniques. Transition Path Sampling (TPS) is a well-established method for generating reactive trajectories, which is simple to implement and does not require the definition of a preconceived reaction coordinate. However, its efficiency is limited by its sequential nature and the resulting correlations between sampled paths. Previous work addressed this limitation by combining TPS with a sampling scheme based on conditioned Boltzmann Generators, a generative machine learning model capable of sampling a given target probability distribution. This approach produces uncorrelated transition paths but relies on an accurate reaction coordinate, which is rarely known in advance. Building on recent advances in committor learning, specifically on the Artificial Intelligence for Molecular Mechanism Discovery (AIMMD) method, in this work we introduce GenAIMMD, an iterative algorithm that actively and self-consistently learns the ideal reaction coordinate (the committor) and trains a conditioned Boltzmann Generator to sample from arbitrary bias windows along it. GenAIMMD thereby provides a correlation-free and fully parallelizable path sampling scheme that does not require prior knowledge of the system's transition mechanism. We apply GenAIMMD to a two-dimensional toy model and a higher-dimensional polymer system. In both cases, GenAIMMD succeeds in training the Boltzmann Generator and learning the committor. Benchmark results show a substantial increase in performance compared to standard TPS.

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