The raw input data as given by a vector is transformed into a piece-wise constant function (a). Figure 2: Sketch of the masking or time-multiplexing procedure. Optical and electro-optical systems consisting of only a single node with delay have been successfully used for reservoir computing, and are especially suited due to their high speeds. Similarly, lasers can also be delay-coupled Soriano et al. Here, the emitted electromagnetic waves can be described with the help of a delay term. The most common example are laser systems, where a laser with delayed self-feedback via a mirror has been studied extensively in the literature Lang and Kobayashi ( 1980) Alsing et al. Many systems in nature can be described by systems of DDEs, where the delay-term usually hides a compressed spatial variable. Mathematically the phase space dimension of a DDE system is infinite. These DDEs contain terms that are not only dependent on the instantaneous variables X ( t ) ∈ R N, but also on their delayed states before a certain time X ( t − τ ). ( 2010), which are described by delay-differential equations (DDEs). In fact, training is only applied to the linear output weighting, for which a simple linear regression is enough to find the optimal values Jaeger ( 2001).Ī class of systems that is naturally suited for reservoir computing are delay systems Appeltant et al. While conventional deep convolutional neural network learning schemes heavily focus on the training of the internal degrees of the network, the ’reservoir’ is assumed to be fixed for reservoir computing. The high dimensional response of the dynamical reservoir is then read out and used as the basis for reconstructing the desired output. This process is often called ’expansion in feature space’, as the resulting trajectory can be of a much higher dimension than the original data series. The dynamical system will then be driven by input data, resulting in some trajectory in its phase space. input light pulses into an optical system. the driving current of a laser or the voltage applied to neurons, or injected with a driving signal, e.g. The data is fed into the system via some number of parameters, e.g. Historically, these systems were first envisioned to be networks of discrete maps Jaeger ( 2001) or neural models Maass et al. At the core of the reservoir computer lies a dynamical system with a high phase-space, also called ’reservoir’. ( 2013) proposes to analytically calculate the response and time series of such a virtual network and then use the analytic formula to speed up computation.įigure 1 depicts a sketch of the reservoir computing paradigm. ( 2014) or the use of counter-propagating fields in a ring laser for simultaneous tasks Nguimdo et al. Possible extensions to the virtual network scheme have also been considered, among others are hierarchical time-multiplexing Zhang et al. ( 2017) and opto-electronic Larger et al. Several groups have successfully implented such a delay-line based reservoir computer using both optic Brunner et al. Instead of an extended physical system or large network of single units, this virtual network approach uses a long delay-line to produce a high dimensional phase-space in time. presented a novel scheme Appeltant et al. Interest in reservoir computing was renewed especially in the photonics and semiconductor community, after Appeltant et al.
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