Learning moment closure in reaction-diffusion systems with spatial dynamic Boltzmann distributions

Oliver K. Ernst, Thomas M. Bartol, Terrence J. Sejnowski, and Eric Mjolsness
Phys. Rev. E 99, 063315 – Published 26 June 2019

Abstract

Many physical systems are described by probability distributions that evolve in both time and space. Modeling these systems is often challenging due to their large state space and analytically intractable or computationally expensive dynamics. To address these problems, we study a machine-learning approach to model reduction based on the Boltzmann machine. Given the form of the reduced model Boltzmann distribution, we introduce an autonomous differential equation system for the interactions appearing in the energy function. The reduced model can treat systems in continuous space (described by continuous random variables), for which we formulate a variational learning problem using the adjoint method to determine the right-hand sides of the differential equations. This approach can be used to enforce a reduced physical model by a suitable parametrization of the differential equations. The parametrization we employ uses the basis functions from finite-element methods, which can be used to model any physical system. One application domain for such physics-informed learning algorithms is to modeling reaction-diffusion systems. We study a lattice version of the Rössler chaotic oscillator, which illustrates the accuracy of the moment closure approximation made by the method and its dimensionality reduction power.

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  • Received 22 April 2019

DOI:https://doi.org/10.1103/PhysRevE.99.063315

©2019 American Physical Society

Physics Subject Headings (PhySH)

Physics of Living SystemsStatistical Physics & ThermodynamicsInterdisciplinary Physics

Authors & Affiliations

Oliver K. Ernst

  • Department of Physics, University of California at San Diego, La Jolla, California 92093, USA

Thomas M. Bartol1

  • Salk Institute for Biological Studies, La Jolla, California 92037, USA

Terrence J. Sejnowski

  • Salk Institute for Biological Studies, La Jolla, California 92037, USA and Division of Biological Sciences, University of California at San Diego, La Jolla, California 92093, USA

Eric Mjolsness

  • Departments of Computer Science and Mathematics, and Institute for Genomics and Bioinformatics, University of California at Irvine, Irvine, 92697 California, USA

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Issue

Vol. 99, Iss. 6 — June 2019

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