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

Abstract

Echo State Networks (ESNs) are a class of single-layer recurrent neural networks with randomly generated internal weights, and a single layer of tuneable outer weights, which are usually trained by regularised linear least squares regression. Remarkably, ESNs still enjoy the universal approximation property despite the training procedure being entirely linear. In this paper, we prove that an ESN trained on a sequence of scalar observations from an ergodic dynamical system (with invariant measure {\mu}) using Tikhonov least squares will approximate future observations of the dynamical system in the L2({\mu}) norm. We call this the ESN Training Theorem. We demonstrate the theory numerically by training an ESN using Tikhonov least squares on a sequence of scalar observations of the Lorenz system, and compare the invariant measure of these observations with the invariant measure of the future predictions of the autonomous ESN.