Autonomous Flows: Exact Finite Interpolation and Uniform Approximation Obstructions
Liu Q, Lü Q, Zuazua E (2026)
Publication Language: English
Publication Status: Submitted
Publication Type: Unpublished / Preprint
Future Publication Type: Article in Edited Volumes
Publication year: 2026
URI: https://dcn.nat.fau.eu/wp-content/uploads/ANODE_2026.pdf
Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/ANODE_2026.pdf
Abstract
In dimension at least two, a class of locally Lipschitz vector fields realizes every finite correspondence
between distinct inputs and distinct targets at any prescribed positive time, provided that it is linear, its
members generate global flows, and it approximates every smooth vector field of bounded support uniformly
on bounded sets. The input and target sets may overlap. The proof constructs a smooth autonomous reference
flow and finitely many localized correction fields, then uses uniform stability and Brouwer degree to obtain
exact endpoints within the approximating class. The argument uses only uniform approximation of the vector
fields; it does not require control of their derivatives. Applied to shallow ReLU vector fields, the result
gives exact finite interpolation without time-dependent coefficients or additional state variables, together with
bounds on width and normalized coefficient strength. In contrast, no continuous autonomous semiflow can
exchange and compress two disjoint balls. Its time maps therefore fail to approximate all continuous maps
uniformly on compact sets. The results distinguish exact interpolation at finitely many points from uniform
control of neighborhoods.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
China (CN)How to cite
APA:
Liu, Q., Lü, Q., & Zuazua, E. (2026). Autonomous Flows: Exact Finite Interpolation and Uniform Approximation Obstructions. (Unpublished, Submitted).
MLA:
Liu, Qiaomu, Qi Lü, and Enrique Zuazua. Autonomous Flows: Exact Finite Interpolation and Uniform Approximation Obstructions. Unpublished, Submitted. 2026.
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