arXiv:2602.08640v5 Announce Type: replace-cross 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 $