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

Abstract

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. Thus, our algorithm incorporates the local geometry and density at each point into a global picture that merges data from different simulation runs in a natural way. Furthermore, we describe lifting and restriction operators between the diffusion map space and the original space. These operators facilitate the description of the coarse-grained dynamics, possibly in the form of a low dimensional effective free energy surface parameterized by the diffusion map reduction coordinates. They also enable a systematic exploration of such effective free energy surfaces through the design of additional “intelligently biased” computational experiments. We conclude by demonstrating our method in a few examples.