Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry, and biology. In this paper we use the first few eigenfunctions of the backward Fokker–Planck diffusion operator as a coarse-grained low dimensional representation for the long-term evolution of a stochastic system and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational data-driven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph...