Reservoir computing with the Kuramoto model

Abstract

Reservoir computing aims to achieve high-performance and low-cost machine learning with a dynamical system as a reservoir. However, in general, there are almost no theoretical guidelines for its high-performance or optimality. Therefore, this paper aims to propose the new concept {\it the edge of bifurcation} for designing a high-performance reservoir, and provide a mathematical justification for it. This concept is a generalization of the famous criterion {\it the edge of chaos}. For this purpose, this paper focuses on the reservoir computing with the Kuramoto model and theoretically reveals its approximation ability. The main result provides an explicit expression of the dynamics of the Kuramoto reservoir by using the order parameters. Thus, the output of the reservoir computing is expressed as a linear combination of the order parameters. As a corollary, sufficient conditions on hyperparameters are obtained so that the set of the order parameters gives the complete basis of the Lebesgue space. This implies that the Kuramoto reservoir has a universal approximation property. Furthermore, the edge of bifurcation is also discussed from the viewpoint of its approximation ability. It is numerically demonstrated by prediction tasks.