Learning the temporal evolution of multivariate densities via normalizing flows

• URL: http://arxiv.org/abs/2107.13735v1
• Date: Thu, 29 Jul 2021 04:05:02 GMT
• Title: Learning the temporal evolution of multivariate densities via normalizing flows
• Authors: Yubin Lu, Romit Maulik, Ting Gao, Felix Dietrich, Ioannis G. Kevrekidis, Jinqiao Duan
• Abstract summary: We propose a method to learn probability distributions using sample path data from differential equations. We analyze this evolution through machine learning assisted construction of a time-dependent mapping. We demonstrate that this approach can learn solutions to non-local Fokker-Planck equations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we propose a method to learn 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 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 learn solutions to non-local Fokker-Planck equations, such as those arising in 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.

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