# Universal approximation of flows of control systems by recurrent neural networks

arXiv: https://arxiv.org/abs/2304.00352 · HTML: https://mlsanity.com/paper/2304.00352/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2023.

## Built on by (most cited first, 2 of 2)

- [Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling](https://mlsanity.com/paper/2602.16864/) (2026)
- [RHYME-XT: A Neural Operator for Spatiotemporal Control Systems](https://mlsanity.com/paper/2603.17867/) (2026)

## Abstract (arXiv)

We consider the problem of approximating flow functions of continuous-time dynamical systems with inputs. It is well-known that continuous-time recurrent neural networks are universal approximators of this type of system. In this paper, we prove that an architecture based on discrete-time recurrent neural networks universally approximates flows of continuous-time dynamical systems with inputs. The required assumptions are shown to hold for systems whose dynamics are well-behaved ordinary differential equations and with practically relevant classes of input signals. This enables the use of off-the-shelf solutions for learning such flow functions in continuous-time from sampled trajectory data.