Chaotic attractor reconstruction using small reservoirs -- the influence of topology

Abstract

Forecasting timeseries based upon measured data is needed in a wide range of applications and has been the subject of extensive research. A particularly challenging task is the forecasting of timeseries generated by chaotic dynamics. In recent years reservoir computing has been shown to be an effective method of forecasting chaotic dynamics and reconstructing chaotic attractors from data. In this work strides are made toward smaller and lower complexity reservoirs with the goal of improved hardware implementability and more reliable production of adequate surrogate models. We show that a reservoir of uncoupled nodes more reliably produces long term timeseries predictions than complex reservoir topologies. We then link the improved attractor reconstruction of the uncoupled reservoir with smaller spectral radii of the resulting surrogate systems. These results indicate that, the node degree plays an important role in determining whether the desired dynamics will be stable in the autonomous surrogate system which is attained via closed-loop operation of the trained reservoir. In terms of hardware implementability, uncoupled nodes would allow for greater freedom in the hardware architecture because no complex coupling setups are needed and because, for uncoupled nodes, the system response is equivalent for space and time multiplexing.

Figures (9)

• Figure 1: (a) Sketch of the reservoir computing approach. An input is fed into the reservoir with fixed random weights ${ \bf W}_{in}$ (grey arrows) and the output is produced by linearly combining states of the reservoir with trained output weights ${\bf W}$ (white arrows). (b) In open-loop operation externally sourced input data is fed into the reservoir to produce the output sequence. (c) In closed-loop operation the trained system is run autonomously by feeding the output back in as the next input step.
• Figure 2: Reservoir network topologies corresponding to the coupling matrices ${\bf \tilde{W}}_{rand}$ (Random), ${\bf \tilde{W}}_{ring}$ (Ring) and ${\bf \tilde{W}}_{id}$ (Uncoupled).
• Figure 3: NRMSE of the $X$, $Y$ and $Z$ components for one-step-ahead prediction in open-loop operation as a function of spectral radius $\rho$ of the coupling matrix. The mean over 100 realisations of the random weights and Lorenz trajectories is depicted for the uncoupled (teal), random (orange) and ring (grey) reservoirs. The error bars indicate the standard deviation.
• Figure 4: Percentage of the 100 realisations of the random weights and Lorenz trajectories for which the resulting closed-loop (autonomous) trajectory; (a) was bounded by $|X|,|Y|,|Z|<2$ for the entire prediction phase of 5000 steps, (b) was bounded and remained oscillatory for the entire prediction phase of 5000 steps.
• Figure 5: Closed-loop prediction performance: (a)-(c) The VPT for the three reservoir topologies. (d)-(f) The attractor deviation ADev, with a reference ADev calculated for two true Lorenz trajectories indicated by the dashed line. (g)-(i) The time interval over which the trajectories were oscillatory. For the ADev and run time values only the bounded trajectories were considered. For all plots the median is plotted and the shaded regions show the 25$^\textrm{th}$ to 75$^\textrm{th}$ percentiles.