Inverse problems for dynamic patterns in coupled oscillator networks: When larger networks are simpler

TL;DR

This work addresses inferring high-dimensional parameters in networks of coupled phase oscillators from observed dynamic patterns, such as chimera states. It develops a noninvasive reconstruction framework based on statistical equilibrium relations (SERs) derived from mean-field/Ott-Antonsen analysis, linking time-averaged observables $\Omega_j$ and $\zeta_j$ to model parameters and the coupling function $G(x)$. The approach provides explicit SERs for the Kuramoto-Battogtokh nonlocal ring, implements a practical reconstruction pipeline (linear regression for $\omega$ and $\beta$, Fourier expansion for $G(x)$ with a tailored minimization $J_{\mathrm{incoh}}(\beta)$), and demonstrates robustness to finite $N$, uneven sampling, and partial data across multiple coupling types. This framework offers a scalable, data-efficient pathway for inverse problems in large oscillator networks and has potential extensions to other topologies, higher-order interactions, and neural or power-grid-inspired systems.

Abstract

Networks of coupled phase oscillators are one of the most studied dynamical systems with numerous applications in physics, chemistry, biology, and engineering. Their behaviour is often characterized by the emergence of various partially synchronized dynamic pattern, which in the case of large networks can be analysed using a variant of the mean-field approach. This method allows to predict what type of network dynamics can be observed for different system parameters. But it is less known that for different partially synchronized patterns it also allows to obtain statistical equilibrium relations that express the dependence of some time-averaged observable quantities of individual oscillators on the internal parameters of these oscillators and the interaction functions between them. In this paper, we show how such relations can be derived, what their typical accuracy is for finite-size networks, and how they can be used to reconstruct the parameters of the corresponding model. The proposed method is particularly effective for large networks, for unevenly sampled or noisy observables, and for partial observations. Its possibilities are demonstrated by application to chimera states in networks of phase oscillator with nonlocal coupling. The extension of the method to other systems with all-to-all and random network topologies is also described.

Figures (11)

• Figure 1: Schematic representation of the proposed parameter reconstruction method. Given a complex spatio-temporal pattern in a system of coupled phase oscillators, the model parameters can be reconstructed by calculating a small number of averages and using statistical equilibrium relations relevant to this model.
• Figure 2: Time-averaged characteristics of chimera states. A typical chimera state in system (\ref{['Eq:Oscillators']}) for a top-hat coupling function with $\sigma = 0.7$, $\omega = 1$, $\alpha = \pi/2 - 0.1$, and $N = 1024$. (a) Space-time plot of $\theta_j(t)$. (b), (c) Effective frequencies $\Omega_j$ and local order parameters $\zeta_j$ obtained by averaging over $2000$ time units. Every 16th point $x_j$ is shown.
• Figure 3: Accuracy of statistical equilibrium relations for finite $N$. Mean discrepancies of SERs (\ref{['SER:1']})--(\ref{['SER:3']}) for the chimera state from Fig. \ref{['Fig:Chimera']}. The discrepancies $\delta_1$, $\delta_2$ and $\delta_3$ defined in the text are shown separately in panels (a), (b) and (c) respectively. The five curves in each panel show the results for different system sizes $N$ mentioned in panel (a).
• Figure 4: Reconstruction of model parameters using statistical equilibrium relations (SER). (a) SER (\ref{['SER:1']}) for the chimera state in Fig. \ref{['Fig:Chimera']}. The circles show the averages calculated from the numerical trajectory (only every 16th point is shown), the line shows a linear fit. (b) The graph of the function $J_\mathrm{incoh}(\beta)$. The dashed line shows the position of the minimum $\beta_\mathrm{min}$. (c) The red/dark curve shows the reconstructed coupling function with $M = 10$ spatial Fourier modes. The grey/light curve shows the original coupling function $G(x)$ and the dotted curve shows its exact Fourier approximation with $10$ modes.
• Figure 5: Efficiency of the proposed method in reconstructing the phase lag $\beta$. The method was applied to the chimera state observed in model (\ref{['Eq:Oscillators']}) with top-hat coupling. The dots show the values of $\beta_\mathrm{min}$ found as the global minimum of the function $J_\mathrm{incoh}(\beta)$ for different $\beta$. Parameters: $N = 2048$, $\omega = 1$ and $\sigma = 0.7$.