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

TL;DR

This paper proves that Neural ODEs can approximate complex dynamical systems with multistability and limit cycles over an infinite time horizon, providing a new theoretical foundation for long-term modeling.

Contribution

It introduces the first universal approximation theorems for Neural ODEs on multistable systems with infinite-time guarantees, extending prior finite-horizon results.

Findings

Neural ODEs approximate Morse-Smale systems with hyperbolic fixed points.

Neural ODEs achieve approximation for systems with hyperbolic limit cycles.

A temporal generalization bound links trajectory closeness to training error.

Abstract

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 $\varepsilon$-$\delta$ closeness -- trajectories within error $\varepsilon$ except for initial conditions of measure $< \delta$ -- over the \emph{infinite} time horizon $[0,\infty)$ 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: $\varepsilon$-$\delta$ closeness implies $L^p$ error $\leq \varepsilon^p + \delta \cdot D^p$ for all $t \geq 0$, bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics.