  One of the most influential results in neural network theory is the universal
approximation theorem [1, 2, 3] which states that continuous functions can be
approximated to within arbitrary accuracy by single-hidden-layer feedforward
neural networks. The purpose of this paper is to establish a result in this
spirit for the approximation of general discrete-time linear dynamical systems
- including time-varying systems - by recurrent neural networks (RNNs). For the
subclass of linear time-invariant (LTI) systems, we devise a quantitative
version of this statement. Specifically, measuring the complexity of the
considered class of LTI systems through metric entropy according to [4], we
show that RNNs can optimally learn - or identify in system-theory parlance -
stable LTI systems. For LTI systems whose input-output relation is
characterized through a difference equation, this means that RNNs can learn the
difference equation from input-output traces in a metric-entropy optimal
manner.
