Learning the temporal evolution of multivariate densities via normalizing flows

Author:

Lu Yubin1ORCID,Maulik Romit23ORCID,Gao Ting1,Dietrich Felix4ORCID,Kevrekidis Ioannis G.5ORCID,Duan Jinqiao3ORCID

Affiliation:

1. School of Mathematics and Statistics and Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China

2. Argonne Leadership Computing Facility, Argonne National Laboratory, Lemont, Illinois 60439, USA

3. Department of Applied Mathematics, College of Computing, Illinois Institute of Technology, Chicago, Illinois 60616, USA

4. Department of Informatics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. Munich, Germany

5. Departments of Applied Mathematics and Statistics and Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21211, USA

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 Lévy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.

Funder

U.S. Department of Energy

Army Research Office

Argonne Leadership Computing Facility

Publisher

AIP Publishing

Subject

Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics

Reference65 articles.

1. X. Bacon, “Optimal transportation of vector-valued measures,” arXiv:1901.04765 (2019).

2. C. Beck, S. Becker, P. Grohs, N. Jaafari, and A. Jentzen, “Solving stochastic differential equations and Kolmogorov equations by means of deep learning,” arXiv:1806.00421 (2018).

3. G. J. Both and R. Kusters, “Temporal normalizing flows,” arXiv:1912.09092v1 (2019).

4. Discovering governing equations from data by sparse identification of nonlinear dynamical systems

5. Data-driven discovery of coordinates and governing equations

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