Reservoir computing with the Kuramoto model

TL;DR

This paper introduces the 'edge of bifurcation' concept for designing high-performance reservoir computing systems, using the Kuramoto model to theoretically justify and demonstrate its universal approximation capabilities.

Contribution

It generalizes the 'edge of chaos' criterion, providing a mathematical framework and explicit dynamics for Kuramoto-based reservoirs with proven universal approximation.

Findings

Kuramoto reservoir has universal approximation property.

Explicit dynamics expressed via order parameters.

Edge of bifurcation enhances reservoir performance.

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.