Learning the temporal evolution of multivariate densities via normalizing flows
This work addresses the challenge of modeling dynamic probability densities in stochastic systems, which is incremental as it applies an existing method (normalizing flows) to a new context of temporal evolution.
The paper tackles the problem of learning temporally evolving multivariate probability distributions from sample path data of stochastic differential equations, using normalizing flows to construct time-dependent mappings from a reference distribution to each snapshot, and demonstrates its effectiveness on systems driven by Brownian and Lévy noise in 2D and 3D examples.
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 Lévy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.