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One of the most influential results in neural network theory is the universalapproximation theorem [1, 2, 3] which states that continuous functions can beapproximated to within arbitrary accuracy by single-hidden-layer feedforwardneural networks. The purpose of this paper is to establish a result in thisspirit for the approximation of general discrete-time linear dynamical systems- including time-varying systems - by recurrent neural networks (RNNs). For thesubclass of linear time-invariant (LTI) systems, we devise a quantitativeversion of this statement. Specifically, measuring the complexity of theconsidered class of LTI systems through metric entropy according to [4], weshow that RNNs can optimally learn - or identify in system-theory parlance -stable LTI systems. For LTI systems whose input-output relation ischaracterized through a difference equation, this means that RNNs can learn thedifference equation from input-output traces in a metric-entropy optimalmanner.

Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems