An Echo State Network Imparts a Curve Fitting

Recurrent neural networks (RNNs) are successfully employed in processing information from temporal data. Approaches to training such networks are varied and reservoir computing-based attainments, such as the echo state network (ESN), provide great ease in training. Akin to many machine learning algorithms rendering an interpolation function or fitting a curve, we observe that a driven system, such as an RNN, renders a continuous curve fitting if and only if it satisfies the echo state property. The domain of the learned curve is an abstract space of the left-infinite sequence of inputs and the codomain is the space of readout values. When the input originates from discrete-time dynamical systems, we find theoretical conditions under which a topological conjugacy between the input and reservoir dynamics can exist and present some numerical results relating the linearity in the reservoir to the forecasting abilities of the ESNs.