Reservoir computing with the Kuramoto model

TL;DR

The paper addresses principled design of reservoir computing by analyzing a Kuramoto-model reservoir in the mean-field limit, deriving a representation $y(t)=\\sum_n w_n r_n(\\alpha u;t)$ and an explicit form for $r_n(\\alpha u;t)$. It proves a completeness condition for the family $\\{r_n\\}$ in $L^2_{\\mathrm{per}}$, yielding universal approximation when $\\Omega\\neq 0$ and $|\\alpha|$ is small, and it develops an edge-of-bifurcation perspective showing Hopf-type bifurcations near $K_c$ enhance performance. The work connects theory and numerics by showing that, near the edge of bifurcation, the Kuramoto reservoir provides accurate time-series predictions for almost periodic inputs and robust autonomous operation. These results offer principled criteria for designing low-cost, highly capable reservoirs based on synchronization transitions.

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.

Figures (7)

• Figure 1: Schematic representation of oscillators on a circle. The left represents a de-synchronous state with $|r|\approx 0$. The right represents a synchronous state with $|r|\approx 1$.
• Figure 2: Bifurcation diagrams of the order parameter (lower panel) for the cases where (a) $g(\omega)$ is a unimodal Gaussian (b) $g(\omega)$ is a uniform distribution with the average $\Omega$. The bifurcation diagrams are independent of $\Omega$ as was mentioned in the previous Remark. Note that for (b), the order parameter takes a large value just after the bifurcation.
• Figure 3: The Kuramoto reservoir computing system. (a) Schematic diagram of the Kuramoto reservoir computing model and the prediction task. (b) The relationship of the order parameter and the bifurcation with coupling strength $K$ when $g(\omega)$ is Gaussian distribution or uniform distribution in the numerical simulation. The graphs indicate the mean of 10 trials; the shade indicates the 95% confidence interval.
• Figure 4: Prediction task using the Van der Pol oscillator. (a) The output $y(t)$ (blue line) and the target function $y_{\mathrm{tar}}(t)$ (orange line) when $\Omega=0.5$ with $K=0.60$ ($<K_c$), $K=0.63$ ($<K_c$), $K=0.64$ ($>K_c$) and $K=0.70$ ($>K_c$), for which we used the Gaussian distribution, and the bifurcation point is $K_c \sim 0.638$. The $y_{\mathrm{tar}}(t)$ is Eq.\ref{['vanderpol']}, with $\mu \sim 5.66$. The training period is shown in shading. (b) The relationship between the mean squared error and the coupling strength $K$ when $\Omega =0$ and 0.5. The solid line represents the mean of 10 trials; the shaded area indicates the 95% confidence interval. (c) Comparison of Gaussian and uniform distribution of $g(\omega)$, with setting $\Omega$ to 0.5.
• Figure 5: Prediction task using an almost periodic function. (a) The output $y(t)$ (green line) and the target function $y_{\mathrm{tar}}(t)$ (orange line) with $\Omega=0.2$ for coupling strengths: $K=0.63$ ($<K_c$) and $K=0.65$ ($>K_c$), where $g(\omega)$ is uniform distribution. Using Gaussian distribution yields the similar result. (b) The mean squared error ($n = 10$).

Theorems & Definitions (23)

• Remark 2.1
• Theorem 4.1
• Remark 4.2
• Corollary 4.3
• Lemma 4.4
• proof : Proof of Theorem \ref{['thm:main']}
• Lemma 4.5
• proof : Proof of Corollary \ref{['cor:main']}
• Theorem 4.6
• Corollary 4.7: edge of bifurcation