# Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees

arXiv: https://arxiv.org/abs/2602.08640 · HTML: https://mlsanity.com/paper/2602.08640/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2026.

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

- [Continuous-Time Piecewise-Linear Recurrent Neural Networks](https://mlsanity.com/paper/2602.15649/) (2026)
- [Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies](https://mlsanity.com/paper/2608.10738/) (2026)

## Abstract (arXiv)

Universal approximation theorems establish the expressive capacity of neural network architectures. For dynamical systems, existing results are limited to finite time horizons or systems with a globally stable equilibrium, leaving multistability and limit cycles unaddressed. We prove that Neural ODEs achieve varepsilon-δ closeness -- trajectories within error varepsilon except for initial conditions of measure < δ -- over the infinite time horizon [0,∞) for three target classes: (1) Morse-Smale systems (a structurally stable class) with hyperbolic fixed points, (2) Morse-Smale systems with hyperbolic limit cycles via exact period matching, and (3) systems with normally hyperbolic continuous attractors via discretization. We further establish a temporal generalization bound: varepsilon-δ closeness implies Lp error ≤ varepsilonp + δ · Dp for all t ≥ 0, bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics.