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

Area of Science:

• Dynamical Systems Theory
• Machine Learning
• Neural Networks

Background:

• Universal approximation theorems demonstrate neural network expressive power.
• Existing theorems for dynamical systems are limited to finite time or globally stable equilibria.
• Multistability and limit cycles in infinite-time dynamical systems remain unaddressed by current theories.

Purpose of the Study:

• To extend universal approximation capabilities of Neural ODEs to infinite time horizons.
• To address the approximation of multistable dynamical systems, including those with limit cycles.
• To bridge theoretical guarantees with practical training metrics for neural network dynamics.

Main Methods:

• Proved $\varepsilon$-$δ$ closeness for Neural ODEs over the infinite time horizon $[0,\infty)$.
• Applied methods to three classes: Morse-Smale systems with hyperbolic fixed points, Morse-Smale systems with hyperbolic limit cycles (using period matching), and systems with normally hyperbolic attractors (via discretization).
• Established a temporal generalization bound linking $\varepsilon$-$δ$ closeness to $L^p$ error.

Main Results:

• Neural ODEs achieve $\varepsilon$-$δ$ closeness for infinite-horizon dynamics in specified classes.
• Demonstrated approximation of systems with multistability and limit cycles.
• Derived a bound showing $\varepsilon$-$δ$ closeness implies bounded $L^p$ error for all $t \geq 0$.

Conclusions:

• This work provides the first universal approximation framework for multistable infinite-horizon dynamics using Neural ODEs.
• The findings extend the theoretical understanding of neural network capabilities for complex systems.
• The temporal generalization bound connects theoretical approximation quality to practical error metrics.