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

2026-05-04

AI Summary

This work addresses the inefficiency of sampling high-dimensional multimodal Gibbs measures, a challenge exacerbated by the curse of dimensionality in collective variable (CV) space that limits conventional enhanced sampling methods. The authors propose a path-dependent regularized McKean–Vlasov dynamics framework that directly regularizes the marginal density of CVs and incorporates a history-weighted empirical measure to enhance statistical stability with limited samples. By employing a functional hierarchical tensor decomposition—free from optimization—to efficiently approximate the historical density, the method circumvents costly convolutions over the CV domain. Validated on benchmark potentials and molecular systems, the approach successfully handles enhanced sampling tasks with up to 64-dimensional CVs, significantly improving both efficiency and stability in high-dimensional settings.

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.

Problem

• high-dimensional sampling
• Gibbs measures
• metastable states
• collective variables
• enhanced sampling

Innovation

• path-dependent McKean–Vlasov dynamics
• tensor density approximation
• enhanced sampling
• collective variables
• Wasserstein gradient flow