Local and global perspectives on diffusion maps in the analysis of molecular systems

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, formalising 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). :1901.06936v2 [physics.data-an] arXiv

Unsupervised Learning Methods for Molecular Simulation Data

“…The diffusion map has a direct application to the analysis of molecular dynamics trajectories generated by a diffusion process, as the collective coordinates emerging from it approximate the eigenfunctions of the Fokker–Planck operator of the process. In ref ( 127 ), it has been shown that the diffusion map eigenfunctions are equivalent, up to a constant, to the eigenfunctions of an overdamped Langevin equation.…”

Computing committors in collective variables via Mahalanobis diffusion maps

“…We refer to this approximation as a pointwise approximation of the generator with respect to the dataset. Given a pointwise approximate generator matrix L, we can then pointwise approximate the continuous committor q via our solution to the discrete committor equation (13). In this work, we will show that the Mahalanobis diffusion map (mmap) yields the desired approximation.…”

“…Meshless approaches to solving the committor PDE discretize it to point clouds obtained from MD simulations. The two most promising approaches utilize neural networks [10,11,12] and/or diffusion maps [13]. The approach based on neural networks is more straightforward in its implementation, while the one based on diffusion maps is more interpretable, visual, and intuitive, and we will focus on it in this work.…”

“…The problem of finding the committor then reduces to solving a system of linear algebraic equations of a manageable size. This strategy would be suitable for MD data in the original R 3Na -dimensional space of atomic positions, and has been utilized heavily in previous work [7,13,19,20,21,22]. As mentioned above, biophysicists prefer to keep track of collective variables rather than positions of atoms, as it lowers the dimension and gives more useful information about the molecular configuration.…”

“…Our proof involves only elementary tools such as multivariable calculus and linear algebra. We apply mmap to two common test systems: the alanine dipeptide molecule [7,13,19,25,26,27] and the Lennard-Jones cluster of 7 particles in 2 dimensions (LJ7) [28,29,30]. For both systems, we compute the committor function on trajectory data and provide validation for the results.…”