Constructive interpolation and generalization rates for Neural ODEs: a control perspective

Álvarez López A, Liverani L, Zuazua E (2026)

Publication Language: English

Publication Status: Submitted

Publication Type: Unpublished / Preprint

Future Publication Type: Article in Edited Volumes

Publication year: 2026

Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/alvarezLliveraniZuazua2026constr.pdf

Abstract

We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SANODEs). We constructively prove that SA-NODEs are capable of exact interpolation of admissible finite datasets, and even satisfy a stronger property that we call simultaneous cell controllability (SCC): their flows can map prescribed disjoint cells into arbitrarily small target balls. This property is the mechanism that upgrades interpolation into quantitative generalization, by allowing SA-NODEs to emulate piecewise-constant nonparametric estimators. Consequently, our risk bounds recover the rates of histogram and nearest-neighbor estimators, provided the network width satisfies a conservative scaling with the sample size. Numerical experiments show that trained SA-NODEs achieve competitive—often lower—test errors than these baselines. Finally, we show that the explicit time dependence is essential. Although two-layer autonomous NODEs can interpolate geometrically nondegenerate datasets, structural obstructions prevent them from achieving SCC. These limitations, further confirmed numerically, support the view that SA-NODEs provide a minimal effective architecture for learning.

Authors with CRIS profile

Antonio Álvarez López Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Lorenzo Liverani Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

How to cite

APA:

Álvarez López, A., Liverani, L., & Zuazua, E. (2026). Constructive interpolation and generalization rates for Neural ODEs: a control perspective. (Unpublished, Submitted).

MLA:

Álvarez López, Antonio, Lorenzo Liverani, and Enrique Zuazua. Constructive interpolation and generalization rates for Neural ODEs: a control perspective. Unpublished, Submitted. 2026.

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