This paper is concerned with the fundamental limits of nonlinear dynamical system learning from input-output traces. Specifically, we show that recurrent neural networks (RNNs) are capable of learning nonlinear systems that satisfy a Lipschitz property and forget past inputs fast enough in a metric-entropy optimal manner. As the sets of sequence-to-sequence maps realized by the dynamical systems we consider are significantly more massive than function classes generally considered in deep neural network approximation theory, a refined metric-entropy characterization is needed, namely in terms of order, type, and generalized dimension. We compute these quantities for the classes of exponentially-decaying and polynomially-decaying Lipschitz fading-memory systems and show that RNNs can achieve them.

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