Computing high-dimensional invariant distributions from noisy data

Abstract

The invariant distribution is an important object in the study of randomly perturbed dynamical systems. The existing methods, including traditional finite difference or finite element methods as well as recently developed machine learning-based methods, all require the knowledge of the dynamical equations or adequate equilibrium data for estimating the invariant distribution. In this work, we propose a data-driven method for inferring the dynamics and simultaneously learning the invariant distribution from noisy trajectory data of the dynamical system. The data is not necessarily at equilibrium and may be collected from the transient period of the dynamics. The proposed method combines the idea of maximum likelihood estimation and a decomposition of the force field as suggested by the Fokker-Planck equation. The drift term (or force field), which is in the form of a decomposition with the constraint specified by the Fokker-Planck equation, and the diffusion term are learned from the data using an alternate updating algorithm. The generalized potential, which is the negative logarithm of the invariant distribution multiplied by the noise, is then obtained from the potential part of the decomposition. The proposed method is able to deal with high-dimensional dynamical systems and the small noise regime, without prior knowledge of the dynamical equations. The proposed method is demonstrated by four numerical examples, including a practical 10-dimensional biological network model.