Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems
This provides a theoretical foundation for using ESNs in forecasting applications, but it is incremental as it extends known approximation properties to a specific training method.
The paper proves that Echo State Networks trained with Tikhonov least squares can approximate target functions in the L2(μ) norm for ergodic dynamical systems, and demonstrates this numerically on the Lorenz system for time series forecasting.
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 observations from an ergodic dynamical system (with invariant measure $μ$) using Tikhonov least squares regression against a set of targets, will approximate the target function in the $L^2(μ)$ norm. In the special case that the targets are future observations, the ESN is learning the next step map, which allows time series forecasting. We demonstrate the theory numerically by training an ESN using Tikhonov least squares on a sequence of scalar observations of the Lorenz system.