Consistency properties of chaotic systems driven by time-delayed feedback

Harnessing synthetic active particles for physical reservoir computing

The processing of information is an indispensable property of living systems realized by networks of active processes with enormous complexity. They have inspired many variants of modern machine learning, one of them being reservoir computing, in which stimulating a network of nodes with fading memory enables computations and complex predictions. Reservoirs are implemented on computer hardware, but also on unconventional physical substrates such as mechanical oscillators, spins, or bacteria often summarized as physical reservoir computing. Here we demonstrate physical reservoir computing with a synthetic active microparticle system that self-organizes from an active and passive component into inherently noisy nonlinear dynamical units. The self-organization and dynamical response of the unit are the results of a delayed propulsion of the microswimmer to a passive target. A reservoir of such units with a self-coupling via the delayed response can perform predictive tasks despite the strong noise resulting from the Brownian motion of the microswimmers. To achieve efficient noise suppression, we introduce a special architecture that uses historical reservoir states for output. Our results pave the way for the study of information processing in synthetic self-organized active particle systems.

Reservoir computing based on an external-cavity semiconductor laser with optical feedback modulation

We numerically and experimentally investigate reservoir computing based on a single semiconductor laser with optical feedback modulation. In this scheme, an input signal is injected into a semiconductor laser via intensity or phase modulation of the optical feedback signal. We perform a chaotic time-series prediction task using the reservoir and compare the performances of intensity and phase modulation schemes. Our results indicate that the feedback signal of the phase modulation scheme outperforms that of the intensity modulation scheme. Further, we investigate the performance dependence of reservoir computing on parameter values and observe that the prediction error improves for large injection currents, unlike the results in a semiconductor laser with an optical injection input. The physical origin of the superior performance of the phase modulation scheme is analyzed using external cavity modes obtained from steady-state analysis in the phase space. The analysis indicates that high-dimensional mapping can be achieved from the input signal to the trajectory of the response laser output by using phase modulation of the feedback signal.

Consistency Hierarchy of Reservoir Computers

We study the propagation and distribution of information-carrying signals injected in dynamical systems serving as reservoir computers. Through different combinations of repeated input signals, a multivariate correlation analysis reveals measures known as the consistency spectrum and consistency capacity. These are high-dimensional portraits of the nonlinear functional dependence between input and reservoir state. For multiple inputs, a hierarchy of capacities characterizes the interference of signals from each source. For an individual input, the time-resolved capacities form a profile of the reservoir's nonlinear fading memory. We illustrate this methodology for a range of echo state networks.

Reservoir computing with swarms

We study swarms as dynamical systems for reservoir computing (RC). By example of a modified Reynolds boids model, the specific symmetries and dynamical properties of a swarm are explored with respect to a nonlinear time-series prediction task. Specifically, we seek to extract meaningful information about a predator-like driving signal from the swarm's response to that signal. We find that the naïve implementation of a swarm for computation is very inefficient, as permutation symmetry of the individual agents reduces the computational capacity. To circumvent this, we distinguish between the computational substrate of the swarm and a separate observation layer, in which the swarm's response is measured for use in the task. We demonstrate the implementation of a radial basis-localized observation layer for this task. The behavior of the swarm is characterized by order parameters and measures of consistency and related to the performance of the swarm as a reservoir. The relationship between RC performance and swarm behavior demonstrates that optimal computational properties are obtained near a phase transition regime.

Laminar chaos in nonlinear electronic circuits with delay clock modulation

We study laminar chaos in an electronic experiment. A two-diode nonlinear circuit with delayed feedback shows chaotic dynamics similar to the Mackey-Glass or Ikeda delay systems. Clock modulation of a single delay line leads to a conservative variable delay, which with a second delay line is augmented to dissipative delays, leading to laminar chaotic regimes. We discuss the properties of this particular delay modulation and demonstrate experimental aspects of laminar chaos in terms of power spectra and return maps.

The reservoir's perspective on generalized synchronization

We employ reservoir computing for a reconstruction task in coupled chaotic systems, across a range of dynamical relationships including generalized synchronization. For a drive-response setup, a temporal representation of the synchronized state is discussed as an alternative to the known instantaneous form. The reservoir has access to both representations through its fading memory property, each with advantages in different dynamical regimes. We also extract signatures of the maximal conditional Lyapunov exponent in the performance of variations of the reservoir topology. Moreover, the reservoir model reproduces different levels of consistency where there is no synchronization. In a bidirectional coupling setup, high reconstruction accuracy is achieved despite poor observability and independent of generalized synchronization.

Consistency in echo-state networks

Consistency is an extension to generalized synchronization which quantifies the degree of functional dependency of a driven nonlinear system to its input. We apply this concept to echo-state networks, which are an artificial-neural network version of reservoir computing. Through a replica test, we measure the consistency levels of the high-dimensional response, yielding a comprehensive portrait of the echo-state property.