Data assimilation for networks of coupled oscillators: Inferring unknown model parameters from partial observations

TL;DR

The paper tackles inferring both the states and unknown parameters of networks of coupled oscillators from partial, noisy observations. It introduces an Ensemble Kalman Filter with state-space augmentation and a novel network-specific localization built from the matrix exponential of the adjacency matrix to suppress spurious correlations, enabling accurate estimation of phases and parameters even when only a subset of nodes is observed. The approach is demonstrated on Kuramoto networks and theta-neuron networks across ring and random topologies, showing substantial improvements over standard EnKF, including strong performance with partial observability and in non-synchronizing regimes. The work provides practical guidance on choosing the localization scale, compares alternative localization strategies, and outlines future directions for extending localization to unknown network structures and more complex dynamical couplings, with potential impact on neural and power-grid applications.

Abstract

Inferring the state and unknown parameters of a network of coupled oscillators is of utmost importance. This task is made harder when only partial and noisy observations are available, which is a typical scenario in realistic high-dimensional systems. The general task of inference falls under data assimilation, and a commonly used assimilation method is the Ensemble Kalman Filter. Employing network-specific localization of the forecast covariance, an Ensemble Kalman Filter with state space augmentation is shown to yield highly accurate estimates of both the oscillator phases and unknown model parameters in the case where only a subset of oscillator phases are observed. In contrast, standard data assimilation methods yield poor results. We demonstrate the effectiveness of our approach for Kuramoto oscillators and for networks of theta neurons, using a variety of network topologies.

Figures (13)

• Figure 1: Time evolution of the $N=50$ phases $\phi_i$ of the Kuramoto model (\ref{['eq:Kuramoto']}) showing convergence to a synchronized state. The Kuramoto model parameters are $\kappa = 27$ and $\omega_i \sim \mathcal{N}(0,0.1)$, with a ring network topology $A$ with coupling radius $r=3$, where each node is connected to its $2r$ nearest neighbors.
• Figure 2: Time evolution of the $N=50$ phases $\phi_i$ of the theta neuron model (\ref{['eq:theta_neurons']}) showing a bump state. The theta neuron model parameters are $\kappa = 2$ and $\zeta_i \sim \mathcal{N}(-0.4,0.1)$, with connectivity matrix $B$ described in Section \ref{['sec:DA_theta']}.
• Figure 3: (a) Time-averaged correlation matrix $Q$ for a large ensemble ($M = 10100$), and (b) for a small ensemble ($M=101$), obtained by applying the standard EnKF to the Kuramoto model (\ref{['eq:Kuramoto']}). The time-averaging is performed for a single realization. The augmented state space is $(\phi_{1,\dots,50},\omega_{1,\dots,50})$, hence the block matrix substructure (indicated by red lines) in $Q$ with $\phi$-$\phi$ correlations, $\phi$-$\omega$ correlations, $\omega$-$\phi$ correlations and $\omega$-$\omega$ correlations. Note that the absolute values of the correlations are shown on a logarithmic scale to more clearly illustrate the differences in magnitude between the smaller correlations. The Kuramoto model parameters are as in Fig. \ref{['fig:KM_phi_of_t']}. (c) The localization matrix $\mathcal{L}$ (\ref{['eq:localization_matrix']}) for the associated ring topology using $\lambda = 0.46$ as determined in Section \ref{['sec:choosing_lambda']}.
• Figure 4: Average forecast correlation (averaging $P_k^{\rm f}$ over all nodes and all time steps) as a function of distance $d$ from the node, scaled by the coupling radius $r$, for $r=2,3,5,10$. All results are obtained from runs of the standard EnKF applied to the Kuramoto model ($N=50$, $\kappa=80/r$, $\omega_i\sim \mathcal{N}(0,0.1)$ with a ring topology), with a large ensemble $M=100(2N+1)$. Note that a coupling strength $\kappa = 80/r$ is used to maintain a constant effective coupling strength $\kappa/\langle k \rangle$, where $\langle k \rangle$ is the mean degree, which ensures that the dynamics of the Kuramoto model are equivalent for all $r$.
• Figure 5: Schematic diagram for the condition (\ref{['eq:ring_lambda_1']}) that defines $\lambda$. Shown is the top right corner of the matrix $L$, the gray squares indicate the underlying adjacency matrix (here a ring with $r=2$), the green squares indicate correlations that should be considered significant because they are within a distance of $2r$ from the node, and orange/red squares indicate correlations that should be considered weak and potentially spurious because the distance from the node is greater than $2r$. The orange squares are those whose value will be $\epsilon$ for $\lambda$ satisfying (\ref{['eq:ring_lambda_1']}) with equality, e.g., in the first row $L_{1,2r+2}=\epsilon$.