Brownian Bridge Approximations for Barrier-Crossing Problems

The simulation of rare events is an important problem in chemical physics with numerous challenges and applications. To investigate such phenomena, we study a process known as a generalized Brownian bridge – i.e., a continuous random walk conditioned to lie in a specified region of phase space and/or end in a given region. This random process has broad applicability when one wants to control the endpoint of stochastic systems, which is often the case in fields like polymer physics and reaction systems. However, construction of a bridge requires solving a Backwards Fokker-Planck (BFP) equation which suffers from the “Curse of Dimensionality” and thus is impractical to compute on complex and high dimensional potential energy surfaces. Therefore, we propose leveraging approximate solutions in conjunction with an importance sampling scheme to correct (re-weight) any errors which occur. Specifically, we exploit asymptotic properties of the BFP to generate simple approximate bridges that reach one region before another, and efficiently generate barrier-crossing trajectories in systems with either large potential energy barriers or in a low temperature (low noise) regime. We see that this drastically simplifies the bridge construction while maintaining statistical accuracy.