Conditioned Langevin Dynamics enables efficient sampling of transition paths

TL;DR

This paper introduces a new stochastic method for efficiently generating conditioned Brownian paths using Langevin dynamics, applicable to complex potentials, and provides approximations for different regimes.

Contribution

It presents a novel local SPDE approach to generate conditioned transition paths, with approximations for low temperature and barrier crossing scenarios.

Findings

Method generates statistically independent transition paths.

Approximations are effective in low temperature and barrier crossing regimes.

The approach is computationally efficient and applicable to complex potentials.

Abstract

We propose a novel stochastic method to generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time $t_{f}$ under a given potential $U(x)$. These paths are sampled with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a local Stochastic Partial Differential Equation (SPDE). This equation cannot be solved in general. We present several approximations that are valid either in the low temperature regime or in the presence of barrier crossing. We show that this method warrants the generation of statistically independent transition paths. It is computationally very efficient. We illustrate the method on the two dimensional Mueller potential as well as on the Mexican hat potential.