Abstract
Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism, diffusion maps can accelerate the sampling of the stationary Boltzmann–Gibbs distribution. In this work, we contrast the local and global perspectives on diffusion maps, based on whether or not the data distribution has been fully explored. In the global setting, we use diffusion maps to identify metastable sets and to approximate the corresponding committor functions of transitions between them. We also discuss the use of diffusion maps within the metastable sets, formalizing the locality via the concept of the quasi-stationary distribution and justifying the convergence of diffusion maps within a local equilibrium. This perspective allows us to propose an enhanced sampling algorithm. We demonstrate the practical relevance of these approaches both for simple models and for molecular dynamics problems (alanine dipeptide and deca-alanine).
Highlights
The calculation of thermodynamic averages for complex models is a fundamental challenge in computational chemistry [1], materials modelling [2] and biology [3]
We provide a rigorous perspective on the construction of diffusion maps within a metastable state by formalizing the concept of a local equilibrium based on the quasi-stationary distribution (QSD) [26]
We review the construction of the original diffusion maps and define the target measure diffusion map, which removes some of its limitations
Summary
The calculation of thermodynamic averages for complex models is a fundamental challenge in computational chemistry [1], materials modelling [2] and biology [3]. Many Bayesian statistical inference calculations arising in clustering and classifying datasets, and in the training of artificial neural networks, reduce to sampling a smooth probability distribution in high dimension and are frequently treated using the techniques of statistical physics [12,13] In such systems, a priori knowledge of the CVs is typically not available, so methods that can automatically determine CVs are of high potential value. Diffusion maps [14,15] provide a dimensionality reduction technique which yields a parametrized description of the underlying low-dimensional manifold by computing an approximation of a Fokker–Planck operator on the trajectory point-cloud sampled from a probability distribution (typically the Boltzmann–Gibbs distribution corresponding to prescribed temperature). We conclude by taking up the question of how the QSD can be used as a tool for the enhanced sampling of large-scale molecular models, paving the way for a full implementation of the described methodology in software framework
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