Designing learning in high dimensional oscillator networks with low dimensional read-out

In this paper we investigate how to configure an oscillator network based reservoir computer with a high number of oscillators and a low dimensional read-out. The read-out is a function on the average phases with respect to each oscillator population. Hence, this read-out provides a robust measurement of the oscillator states. We consider a low number of populations which leads to a low-dimensional read-out. Here, the task is time-series prediction. The input time-series is introduced via a forcing term. After a training phase the input is learned. Importantly, the training weights are introduced in the forcing term meaning that the oscillator network is left untouched. Hence, we can apply classical methods for oscillator networks. Here, we consider the continuum limit for Kuramoto oscillators by using the Ott-Antonsen Ansatz. The success and failure of the reservoir computer is then studied by bifurcations in the coupling and forcing parameter space. We will also show that the average phase read-out can naturally arise when considering the read-out on the phase states. Finally, we give numerical evidence that at least 4 oscillator populations are necessary to learn chaotic target dynamics.