Emergent Spaces for Coupled Oscillators

Frontiers in Computational Neuroscience
Thomas N ThiemIoannis G Kevrekidis

Abstract

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 demonst...Continue Reading

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Citations

May 29, 2020·Journal of Mathematical Neuroscience·Christian BickErik A Martens
Aug 4, 2021·Chaos·Thomas N ThiemIoannis G Kevrekidis

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Software Mentioned

Scipy
Adam
TensorFlow

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