Emergent Spaces for Coupled Oscillators - Thiem, Thomas N.; Kooshkbaghi, Mahdi; Bertalan, Tom; Laing, Carlo R.; Kevrekidis, Ioannis G. (Frontiers in Computational Neuroscience, 2020)
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 order parameters 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 further develop our coarse-graining methodology to enable us to learn evolution equations for the identified coarse variables via a 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 good, 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 the 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. Thus, we have exhibited an integrated suite of tools for obtaining data-driven coarse variables, data-driven effective parameters and data-driven coarse-grained equations from detailed dynamic observations of networks of oscillators. 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 a Chung-Lu type network of Hodgkin-Huxley neurons.