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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi></mml:math> closeness-trajectories within error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> except for initial conditions of measure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo><</mml:mo> <mml:mi>δ</mml:mi></mml:math> -over the <i>infinite</i> time horizon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo></mml:math> 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: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi></mml:math> closeness implies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow> <mml:mrow><mml:mi>p</mml:mi></mml:mrow> </mml:msup> </mml:math> error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>≤</mml:mo> <mml:msup><mml:mrow><mml:mi>ε</mml:mi></mml:mrow> <mml:mrow><mml:mi>p</mml:mi></mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>⋅</mml:mo> <mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow> <mml:mrow><mml:mi>p</mml:mi></mml:mrow> </mml:msup> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn></mml:math> , bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics.