Inferring Coupling Strengths in Synchronized Oscillators

Accurately estimating the coupling strength in oscillator networks from macroscopic observations alone is essential for predicting synchronization transitions. We consider the inverse problem of reconstructing the unknown coupling strength $K$ in the globally coupled Kuramoto model from scalar observations of the macroscopic order parameter $R(t)$, assuming that the natural frequencies and the initial phase configuration are known. This problem is motivated by practical situations in which individual oscillator phases are inaccessible, whereas a coarse-grained collective signal can be measured continuously. Rather than relying on microscopic state observations, our method infers the coupling strength solely from the evolution of the macroscopic order parameter. We employ an extended Kalman filter with an augmented state representation that recursively estimates the coupling strength from observations of $R(t)$. By exploiting the mean-field structure of the globally coupled Kuramoto model, the covariance prediction step can be computed efficiently, substantially reducing the computational cost. Numerical simulations demonstrate that the proposed estimator accurately reconstructs the coupling strength and remains stable even when $R(t)$ is small and strongly fluctuating.