High-Dimensional Enhanced Sampling via Regularized Path-Dependent McKean--Vlasov Dynamics using Tensor Density Approximation

TL;DR

This paper introduces a novel high-dimensional enhanced sampling method using a regularized path-dependent McKean--Vlasov approach with tensor density approximation, improving scalability and stability.

Contribution

It proposes a new regularized, path-dependent McKean--Vlasov formulation for high-dimensional sampling, avoiding convolution over CV space and employing tensor density approximation for scalability.

Findings

Effective sampling for CV dimensions up to 64.

The method improves statistical stability in small-replica regimes.

Numerical experiments validate the approach on benchmark and molecular systems.

Abstract

Sampling from high-dimensional Gibbs measures poses a challenge when the energy landscape consists of multiple metastable states. Enhanced-sampling methods mitigate this difficulty by introducing adaptive biasing potentials to facilitate the exploration along prescribed collective variables (CVs), but their scalability is often limited by the dimension of the CV space. Motivated by the Wasserstein-gradient-flow interpretation of adaptive biasing, we propose a regularized path-dependent McKean--Vlasov formulation for high-dimensional enhanced sampling. The formulation replaces the variational regularization of the Wasserstein functional by a direct regularization of the CV marginal density in the McKean--Vlasov drift, avoiding the outer convolution over the CV domain. Furthermore, it replaces the instantaneous law by a weighted path-history measure to improve statistical stability in the small-replica regime. We establish well-posedness of the resulting regularized and path-dependent stochastic dynamics under suitable assumptions. For numerical realization, the history-averaged CV marginal density is approximated using an optimization-free functional hierarchical tensor representation, leading to a scalable density-based adaptive biasing scheme. Numerical experiments on benchmark potentials and molecular systems demonstrate the effectiveness of the proposed method for sampling problems with CV dimensions up to 64.