Rare events such as conformational changes in biomolecules, phase transitions, and chemical reactions are central to the behavior of many physical systems, yet they are extremely difficult to study computationally because unbiased simulations seldom produce them. Transition Path Theory (TPT) provides a rigorous statistical framework for analyzing such events: it characterizes the ensemble of reactive trajectories between two designated metastable states (reactant and product), and its central object--the committor function, which gives the probability that the system will next reach the product rather than the reactant--encodes all essential kinetic and thermodynamic information. We introduce a framework that casts committor estimation as a stochastic optimal control (SOC) problem. In this formulation the committor defines a feedback control--proportional to the gradient of its logarithm--that actively steers trajectories toward the reactive region, thereby enabling efficient sampling of reactive paths. To solve the resulting hitting-time control problem we develop two complementary objectives: a direct backpropagation loss and a principled off-policy Value Matching loss, for which we establish first-order optimality guarantees. We further address metastability, which can trap controlled trajectories in intermediate basins, by introducing an alternative sampling process that preserves the reactive current while lowering effective energy barriers. On benchmark systems, the framework yields markedly more accurate committor estimates, reaction rates, and equilibrium constants than existing methods.
Figures & tables
Figure 1: Rugged Müller–Brown results, shown as the most metastable overdamped benchmark. Red curves in (a) mark the finite-element q=0.5 transition surface; black curves mark the learned q=0.5 surface. Panels (b,c) compare transition-path ensembles sampled by TPS and by the learned REACT-VM control, while panels (d,e) compare the corresponding hitting-time distributions. Full overdamped committor, path, and flux are shown in Figures 6 , 7 and 8 .
Figure 2Figure 3Figure 4
Figure 5: Ground-truth and learned committors on the periodically-driven triple-well system, overlaid with the time-dependent force field of the system.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Hyperparameter
Triple-well
Müller
Rugged Müller
Model
Network architecture
MLP
Network size
(32, 64, 128, 64, 32)
Training
Learning rate
0.001
Training iterations
20,000
Appendix
Table 5: Hyperparameters for the overdamped Langevin system experiments.
Hyperparameter
Underdamped DW
Maier–Stein
Periodic TW
Model
Network architecture
MLP
Network size
(32, 64, 128, 64, 32)
Training
Learning rate
0.001
Training iterations
40,000
Appendix
Table 6: Hyperparameters for the underdamped and non-reversible experiments.
Figure 6: Ground-truth and learned committors on the three overdamped systems. Red curves mark the finite-element q=0.5 transition surface; black curves mark the q=0.5 surface learned by each method. All REACT-VM visualizations use κ=1.0 .
Figure 7: Transition-path ensembles and hitting-time distributions for TPS and REACT-VM on the three overdamped potentials. Two representative trajectories are highlighted in each path ensemble. REACT-VM results use κ=1.0 for all systems.
Figure 8: Reactive flux visualizations on the three overdamped potentials with κ=0 .
Figure 9: Mean absolute error maps between ground-truth and learned committor functions on the three overdamped systems.
Figure 10: Mean absolute error map between ground-truth and learned committor functions on the underdamped double-well system.
Figure 11: Mean absolute error map between ground-truth and learned committor functions on the Maier–Stein system.
Figure 12: Mean absolute error map between ground-truth and learned committor functions on the periodically-driven triple-well system.
Siebel School of Computing and Data Science, University of Illinois Urbana–Champaign · Department of Electrical and Computer Engineering, University of Illinois Urbana–Champaign
CMU-Pitt Program in Computational Biology Dept. of Computational & Systems Biology University of Pittsburgh · Machine Learning Department Dept. of Mathematical Sciences Carnegie Mellon University · Courant Institute, New York University Machine Learning Lab +1