Data-driven stochastic modeling of coarse-grained dynamics with finite-size effects using Langevin regression

TL;DR

This paper uses Langevin regression to develop stochastic models of the Kuramoto oscillator system, capturing finite-size effects and bifurcation behavior in the order parameter dynamics.

Contribution

It demonstrates that Langevin regression can accurately model finite-size effects in the Kuramoto model, revealing the nature of fluctuations near critical points.

Findings

Diffusion term magnitude scales as N^{-1/2}

Order parameter fluctuations are driven by drift bifurcation

Dynamics are consistent with Ott-Antonsen ansatz in the infinite limit

Abstract

Obtaining coarse-grained models that accurately incorporate finite-size effects is an important open challenge in the study of complex, multi-scale systems. We apply Langevin regression, a recently developed method for finding stochastic differential equation (SDE) descriptions of realistically-sampled time series data, to understand finite-size effects in the Kuramoto model of coupled oscillators. We find that across the entire bifurcation diagram, the dynamics of the Kuramoto order parameter are statistically consistent with an SDE whose drift term has the form predicted by the Ott-Antonsen ansatz in the $N\to \infty$ limit. We find that the diffusion term is nearly independent of the bifurcation parameter, and has a magnitude decaying as $N^{-1/2}$, consistent with the central limit theorem. This shows that the diverging fluctuations of the order parameter near the critical point are driven by a bifurcation in the underlying drift term, rather than increased stochastic forcing.