Emergent Spaces for Coupled Oscillators

Frontiers in Computational Neuroscience

Ioannis G. Kevrekidis¹  Thomas N. Thiem²  Tom Bertalan³  Mahdi Kooshkbaghi⁴  Carlo R. Laing⁵ 

[1] Chemical and Biomolecular Engineering and Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, United States;Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, United States;Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, United States;Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, United States;School of Natural and Computational Sciences, Massey University, Auckland, New Zealand;

关键词: diffusion maps;    manifold learning;    geometric harmonics;    neural networks;    Kuramoto oscillators;    coupled systems;   

DOI  :  10.3389/fncom.2020.00036

来源: DOAJ

【 摘 要 】

Systems of coupled dynamical units (e.g., oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution. Such coarse-graining procedures often require extensive experience and/or a deep understanding of the system dynamics. In this paper we present a systematic, data-driven approach to discovering “bespoke” coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of “effective” parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible extensions of this approach, including the possibility of obtaining data-driven effective partial differential equations for coarse-grained neuronal network behavior, as illustrated by the synchronization dynamics of Hodgkin–Huxley type neurons with a Chung-Lu network. Thus, we build an integrated suite of tools for obtaining data-driven coarse variables, data-driven effective parameters, and data-driven coarse-grained equations from detailed observations of networks of oscillators.

【 授权许可】

Unknown