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.

中文翻译：

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线性动力系统递归神经网络学习的度量熵极限

神经网络理论中最有影响力的结果之一是通用逼近定理[1、2、3]，该定理指出，单隐藏层前馈神经网络可以将连续函数逼近任意精度。本文的目的是在这种精神下建立一个结果，用于通过递归神经网络（RNN）逼近一般离散时间线性动力系统-包括时变系统。对于线性时不变（LTI）系统的子类，我们设计了此声明的定量版本。具体来说，根据[4]通过度量熵来衡量所考虑的LTI系统类别的复杂性，我们表明RNN可以最佳地学习-或以系统理论的话来识别-稳定的LTI系统。