  We leverage data-driven model discovery methods to determine the governing
equations for the emergent behavior of heterogeneous networked dynamical
systems. Specifically, we consider networks of coupled nonlinear oscillators
whose collective behaviour approaches a limit cycle. Stable limit-cycles are of
interest in many biological applications as they model self-sustained
oscillations (e.g. heart beats, chemical oscillations, neurons firing,
circadian rhythm). For systems that display relaxation oscillations, our method
automatically detects boundary (time) layer structures in the dynamics, fitting
inner and outer solutions and matching them in a data-driven manner. We
demonstrate the method on well-studied systems: the Rayleigh Oscillator and the
Van der Pol Oscillator. We then apply the mathematical framework to discovering
low-dimensional dynamics in networks of semi-synchronized Kuramoto, Rayleigh,
Rossler, and Fitzhugh-Nagumo oscillators, as well as heterogeneous combinations
thereof. We also provide a numerical exploration of the dimension of collective
network dynamics as a function of several network parameters, showing that the
discovery of coarse-grained variables and dynamics can be accomplished with the
proposed architecture.
