This is a preliminary version. Markov chain Monte Carlo samplers based on discretizations of (overdamped) Langevin dynamics are commonly used in the Bayesian inference and computational statistical physics literature to estimate high-dimensional integrals. One can introduce a non-constant diffusion matrix to precondition these dynamics, and recent works have optimized it in order to sooner reach stationarity by overcoming entropic and energy barriers. However, the methodology introduced to compute these optimal diffusions is not suited to high-dimensional settings, as it relies on costly optimization procedures. In this work, we propose a class of diffusion matrices, based on one-dimensional collective variables (CVs), which helps dynamics explore the latent space defined by the CV. The form of the diffusion matrix is such that the effective dynamics, which are approximations of the processes as observed on the latent space, are governed by the optimal effective diffusion coefficient in a homogenized limit, which possesses an analytical expression. We describe how this class of diffusion matrices can be constructed and learned during the simulation. We provide implementations of the Metropolis--Adjusted Langevin Algorithm and Riemann Manifold (Generalized) Hamiltonian Monte Carlo algorithms, and discuss numerical optimizations in the case when the CV depends only on a few number of components of the position of the system. We illustrate the efficiency gains of using this class of diffusion by computing mean transition durations between two configurations of a dimer in a solvent.

翻译：本文为初步版本。基于（过阻尼）朗之万动力学离散化的马尔可夫链蒙特卡洛采样器在贝叶斯推断与计算统计物理学文献中广泛用于估计高维积分。通过引入非恒定扩散矩阵可对这些动力学进行预处理，近期研究通过克服熵垒与能量壁垒以加速达到平稳态，从而优化了该矩阵。然而，现有计算此类最优扩散矩阵的方法依赖昂贵的优化流程，难以适用于高维场景。本研究提出一类基于一维集体变量的扩散矩阵，其有助于动力学在集体变量定义的潜空间中探索。该扩散矩阵的形式使得有效动力学（即潜空间观测过程的近似）在均质化极限下受具有解析表达式的最优有效扩散系数支配。我们阐述了此类扩散矩阵的构建方法及其在模拟过程中的学习机制。我们实现了Metropolis-Adjusted Langevin Algorithm与Riemann Manifold (Generalized) Hamiltonian Monte Carlo算法，并针对集体变量仅依赖于系统位置少量分量的情形讨论了数值优化策略。通过计算溶剂中二聚体两种构型间的平均跃迁时长，我们验证了使用此类扩散矩阵带来的效率提升。