Convergence rates for an Adaptive Biasing Potential scheme from a Wasserstein optimization perspective

TL;DR

This paper provides the first exponential convergence analysis for an adaptive biasing potential scheme in molecular simulations, using Wasserstein gradient flow techniques to understand its efficiency.

Contribution

It introduces a novel theoretical framework analyzing convergence of adaptive biasing methods via Wasserstein gradient flows, especially for non-reversible Langevin dynamics.

Findings

First exponential convergence result for non-reversible Langevin diffusion.

Interpretation of the algorithm as gradient descent in probability space.

Theoretical insights into the efficiency of adaptive biasing methods.

Abstract

Free-energy-based adaptive biasing methods, such as Metadynamics, the Adaptive Biasing Force (ABF) and their variants, are enhanced sampling algorithms widely used in molecular simulations. Although their efficiency has been empirically acknowledged for decades, providing theoretical insights via a quantitative convergence analysis is a difficult problem, in particular for the kinetic Langevin diffusion, which is non-reversible and hypocoercive. We obtain the first exponential convergence result for such a process, in an idealized setting where the dynamics can be associated with a mean-field non-linear flow on the space of probability measures. A key of the analysis is the interpretation of the (idealized) algorithm as the gradient descent of a suitable functional over the space of probability distributions.