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Echo State Networks (ESNs) are a class of single-layer recurrent neuralnetworks with randomly generated internal weights, and a single layer oftuneable outer weights, which are usually trained by regularised linear leastsquares regression. Remarkably, ESNs still enjoy the universal approximationproperty despite the training procedure being entirely linear. In this paper,we prove that an ESN trained on a sequence of observations from an ergodicdynamical system (with invariant measure $\mu$) using Tikhonov least squaresregression against a set of targets, will approximate the target function inthe $L^2(\mu)$ norm. In the special case that the targets are futureobservations, the ESN is learning the next step map, which allows time seriesforecasting. We demonstrate the theory numerically by training an ESN usingTikhonov least squares on a sequence of scalar observations of the Lorenzsystem.

Echo State Networks trained by Tikhonov least squares are L2({\mu}) approximators of ergodic dynamical systems