  Many processes in nature such as conformal changes in biomolecules and
clusters of interacting particles, genetic switches, mechanical or
electromechanical oscillators with added noise, and many others are modeled
using stochastic differential equations with small white noise. The study of
rare transitions between metastable states in such systems is of great interest
and importance. The direct simulation of rare transitions is difficult due to
long waiting times. Transition path theory is a mathematical framework for the
quantitative description of rare events. Its crucial component is the committor
function, the solution to a boundary value problem for the backward Kolmogorov
equation. The key fact exploited in this work is that the optimal controller
constructed from the committor leads to the generation of transition
trajectories exclusively. We prove this fact for a broad class of stochastic
differential equations. Moreover, we demonstrate that the committor computed
for a dimensionally reduced system and then lifted to the original phase space
still allows us to construct an effective controller and estimate the
transition rate with reasonable accuracy. Furthermore, we propose an
all-the-way-through scheme for computing the committor via neural networks,
sampling the transition trajectories, and estimating the transition rate
without meshing the space. We apply the proposed methodology to four test
problems: the overdamped Langevin dynamics with Mueller's potential and the
rugged Mueller potential in 10D, the noisy bistable Duffing oscillator, and
Lennard-Jones-7 in 2D.
