# Metric-Entropy Limits on the Approximation of Nonlinear Dynamical Systems

arXiv: https://arxiv.org/abs/2407.01250 · HTML: https://mlsanity.com/paper/2407.01250/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs](https://mlsanity.com/paper/2606.27285/) (2026)

## Abstract (arXiv)

This paper is concerned with fundamental limits on the approximation of nonlinear dynamical systems. Specifically, we show that recurrent neural networks (RNNs) can approximate nonlinear systems -- that satisfy a Lipschitz property and forget past inputs fast enough -- in metric-entropy-optimal manner. As the sets of sequence-to-sequence mappings realized by the dynamical systems we consider are significantly more massive than function classes generally analyzed in 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- and polynomially Lipschitz fading-memory systems and show that RNNs can achieve them.