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.

获得准确结合有限尺寸效应的粗粒度模型是研究复杂的多尺度系统的一个重要的开放挑战。我们应用 Langevin 回归，这是一种最近开发的用于寻找真实采样时间序列数据的随机微分方程 (SDE) 描述的方法，以了解 Kuramoto 耦合振荡器模型中的有限大小效应。我们发现，在整个分叉图中，仓本阶参数的动力学与 SDE 的漂移项具有由 Ott-Antonsen ansatz 预测的形式在统计上一致$N\to \infty$限制。我们发现扩散项几乎与分岔参数无关，并且其幅度衰减为$N^{-1/2}$，与中心极限定理一致。这表明临界点附近阶参数的发散波动是由潜在漂移项中的分叉驱动的，而不是随机强迫的增加。