Exploration of Effective Potential Landscapes using Coarse Reverse Integration

TL;DR

This paper introduces a reverse integration method for efficiently exploring low-dimensional potential landscapes, including in noisy systems, by using coarse ring evolution modes to identify landscape features and transition paths.

Contribution

It presents a novel reverse integration approach that enables effective exploration of potential landscapes in both deterministic and stochastic systems, including equation-free scenarios.

Findings

Successfully identified saddle points and transition paths in known landscapes.

Applied the method to noisy systems with stochastic and molecular dynamics simulators.

Demonstrated the approach's effectiveness in estimating landscape features from short simulation bursts.

Abstract

We describe a reverse integration approach for the exploration of low-dimensional effective potential landscapes. Coarse reverse integration initialized on a ring of coarse states enables efficient "navigation" on the landscape terrain: escape from local effective potential wells, detection of saddle points, and identification of significant transition paths between wells. We consider several distinct ring evolution modes: backward stepping in time, solution arc--length, and effective potential. The performance of these approaches is illustrated for a deterministic problem where the energy landscape is known explicitly. Reverse ring integration is then applied to "noisy" problems where the ring integration routine serves as an outer "wrapper" around a forward-in-time inner simulator. Three versions of such inner simulators are considered: a system of stochastic differential equations, a Gillespie--type stochastic simulator, and a molecular dynamics simulator. In these "equation-free" computational illustrations, estimation techniques are applied to the results of short bursts of "inner" simulation to obtain the unavailable (in closed form) quantities (local drift and diffusion coefficient estimates) required for reverse ring integration; this naturally leads to approximations of the effective landscape.