Learning the temporal evolution of multivariate densities via normalizing flows

TL;DR

This paper introduces a machine learning method using normalizing flows to model the temporal evolution of multivariate probability distributions driven by stochastic differential equations, effectively capturing their dynamics from sample data.

Contribution

It develops a novel approach to learn time-dependent mappings for evolving distributions, extending normalizing flows to dynamic probabilistic systems driven by Fokker-Planck equations.

Findings

Accurately models probability density evolution over time.

Works with systems driven by Brownian and Lévy noise.

Validates with 2D and 3D multimodal distributions.

Abstract

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker-Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a multivariate normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and L\'evy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.