Versatile Reservoir Computing for Heterogeneous Complex Networks

TL;DR

It is demonstrated that a single, small-scale reservoir computer trained on time series from a subset of elements is able to replicate the dynamics of any element in a large-scale complex network, though the elements are of different intrinsic parameters and connectivities.

Abstract

A new machine learning scheme, termed versatile reservoir computing, is proposed for sustaining the dynamics of heterogeneous complex networks. We show that a single, small-scale reservoir computer trained on time series from a subset of elements is able to replicate the dynamics of any element in a large-scale complex network, though the elements are of different intrinsic parameters and connectivities. Furthermore, by substituting failed elements with the trained machine, we demonstrate that the collective dynamics of the network can be preserved accurately over a finite time horizon. The capability and effectiveness of the proposed scheme are validated on three representative network models: a homogeneous complex network of non-identical phase oscillators, a heterogeneous complex network of non-identical phase oscillators, and a heterogeneous complex network of non-identical chaotic oscillators.

Figures (4)

• Figure 1: Sustaining the dynamics of heterogeneous complex networks using versatile RC. (a1) Homogeneous complex network of $N=50$ non-identical phase oscillators (Model A), all with degree $k=3$. Red nodes ($m=10$) denote sampled oscillators used for machine training. (a2) Heterogeneous complex network of $N=50$ non-identical phase oscillators (Model B). The oscillators are of different degrees. Sampled oscillators ($m=10$) are colored by degrees: blue for $k=2$, red for $k=3$, and green for $k=4$. (b) Schematic of the scheme of parallelized RCs for homogeneous networks. (c1) Input data for Model A. (c2) Closed-loop configuration applied to Model A. (d1) Input data for Model B. Triangles represent the virtual oscillators created by the padding technique. (d2) Closed-loop configuration applied to Model B.
• Figure 2: Performance of the machine in substituting a single oscillator in Model A. Shown in (a) are typical results for (a1) non-sampled oscillators with irregular motions, (a2) sampled oscillators with irregular motions, (a3) sampled oscillators with regular motions, and (a4) non-sampled oscillators with regular motions. Top panels: true (grey) vs. predicted (red) state evolutions. Bottom panels: time evolution of the synchronization order parameter, $R(t)$, for the original (grey) and substituted (red) networks. Dashed lines indicate maintenance horizons ($T_s$). (b) Distribution of maintenance horizons across all oscillators. Results for the sampled and non-sampled oscillators are colored in blue and red, respectively.
• Figure 3: Performance of the machine in sustaining the dynamics of Model B. (a1–a4) Same as in Fig. \ref{['fig2']} but for the heterogeneous network. Grey: original system; red: with substitution. (b) Distribution of maintenance horizons for sampled (blue) and non-sampled (red) oscillators.
• Figure 4: Performance of the machine in maintaining the dynamics of Model C. Results for the sampled and non-sampled oscillators are shown in the left and right columns, respectively. In each subplot, the top and bottom panels display, respectively, the state evolution of the substituted oscillator and the time evolution of the network synchronization error, $\delta e$. Grey curves are results for the original network. Red curves are results with substitution. Vertical dashed lines denote the maintenance horizons.