[论文解读] Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees

Abel Sagodi, Il Memming Park|arXiv (Cornell University)|Feb 9, 2026

Neural Networks and Reservoir Computing被引用 0

一句话总结

The paper proves that Neural ODEs can uniformly approximate certain multistable dynamical systems over infinite time horizons, establishing ε-δ guarantees for Morse–Smale systems with hyperbolic fixed points, hyperbolic limit cycles via exact period matching, and normally hyperbolic continuous attractors via discretization, plus a temporal generalization bound.

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 $\varepsilon$-$δ$ 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.

研究动机与目标

• Demonstrate infinite-time universal approximation of multistable dynamical systems by Neural ODEs.
• Extend universal approximation beyond fading memory to Morse–Smale and normally hyperbolic attractors.
• Provide an explicit ε-δ framework linking topological guarantees to training metrics.

提出的方法

• Define ε-δ closeness for infinite-horizon trajectory approximation.
• Use structural stability (Palis–Smale) to bound basin errors near separatrices.
• Apply exact period matching via localized vector field scaling to fix P-type error for limit cycles.
• Employ a tiling strategy to approximate normally hyperbolic continuous attractors by discrete attractors.
• Derive a temporal generalization bound linking ε-δ closeness to time-averaged Lp error.

实验结果

研究问题

• RQ1 Can Neural ODEs achieve ε-δ closeness to target dynamics over [0, ∞) for Morse–Smale systems with hyperbolic fixed points?
• RQ2 Can ε-δ closeness be achieved for Morse–Smale systems with hyperbolic limit cycles through exact period matching?
• RQ3 Can continuous attractors be approximated over infinite time via discretization tiling while controlling D-type error?
• RQ4 What is the relationship between ε-δ trajectory closeness and time-averaged Lp error over infinite horizons?

主要发现

• Neural ODEs can be ε-δ close to Morse–Smale targets with hyperbolic fixed points over the infinite horizon.
• Exact period matching via localized scaling eliminates P-type error for limit cycles, enabling infinite-time approximation.
• A tiling approach allows approximation of normally hyperbolic continuous attractors by discrete tiles with bounded D-type error.
• A temporal generalization bound shows time-averaged Lp error is bounded by ε^p + δ·D^p, linking topology to training metrics.
• The results constitute the first universal approximation framework for multistable infinite-horizon dynamics.