Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems
This provides a theoretical foundation for using RNNs in system identification, though it is incremental as it extends universal approximation ideas to a specific domain.
The paper establishes that recurrent neural networks (RNNs) can optimally learn stable linear time-invariant (LTI) systems, showing they achieve metric-entropy optimal approximation for this class of dynamical systems.
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.