Data-driven discovery of governing equations for coarse-grained heterogeneous network dynamics

TL;DR

This paper introduces a data-driven approach to identify governing equations of complex heterogeneous network dynamics, capturing emergent behaviors like limit cycles and relaxation oscillations in various oscillator networks.

Contribution

The work presents a novel framework for discovering low-dimensional models of heterogeneous networked oscillators, including boundary layer detection and matching, applicable to multiple well-known systems.

Findings

Successfully identified governing equations for various oscillator networks.

Demonstrated automatic detection of boundary layers in relaxation oscillations.

Showed the ability to discover coarse-grained variables in complex networks.

Abstract

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.