Transition path theory for Langevin dynamics on manifold: optimal control and data-driven solver

We present a data-driven point of view for the rare events, which represent conformational transitions in biochemical reactions modeled by over-damped Langevin dynamics on manifolds in high dimensions. Given the point clouds sampled from an unknown reaction dynamics, we construct a discrete Markov process based on an approximated Voronoi tesselation which incorporates both the equilibrium and the manifold information. We reinterpret the transition state theory and transition path theory from the optimal control viewpoint. The controlled random walk on point clouds is utilized to simulate the transition path, which becomes an almost sure event instead of a rare event in the original reaction dynamics. Some numerical examples on sphere and torus are conducted to illustrate the data-driven solver for transition path theory on point clouds. The resulting dominated transition path highly coincides with the mean transition path obtained via the controlled Monte Carlo simulations.