Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910019257106432 |
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| author | Sagodi, Abel Park, Il Memming |
| author_facet | Sagodi, Abel Park, Il Memming |
| contents | 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 \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: $\varepsilon$-$δ$ closeness implies $L^p$ error $\leq \varepsilon^p + δ\cdot D^p$ for all $t \geq 0$, bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08640 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees Sagodi, Abel Park, Il Memming Dynamical Systems Neurons and Cognition 68T07, 37C10 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 \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: $\varepsilon$-$δ$ closeness implies $L^p$ error $\leq \varepsilon^p + δ\cdot D^p$ for all $t \geq 0$, bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics. |
| title | Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees |
| topic | Dynamical Systems Neurons and Cognition 68T07, 37C10 |
| url | https://arxiv.org/abs/2602.08640 |