# Constructive interpolation and generalization rates for neural ODEs: a control perspective

arXiv: https://arxiv.org/abs/2606.00469 · HTML: https://mlsanity.com/paper/2606.00469/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 59 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 30 of 35)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Structure preserving deep learning](https://mlsanity.com/paper/2006.03364/) (2020)
- [Deep neural networks, generic universal interpolation, and controlled ODEs](https://mlsanity.com/paper/1908.07838/) (2019)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Generalization bounds for neural ordinary differential equations and deep residual networks](https://mlsanity.com/paper/2305.06648/) (2023)
- [Interpolation, Approximation and Controllability of Deep Neural Networks](https://mlsanity.com/paper/2309.06015/) (2023)
- [Large-time asymptotics in deep learning](https://mlsanity.com/paper/2008.02491/) (2020)
- [Control in the spaces of ensembles of points](https://mlsanity.com/paper/1907.00905/) (2019)
- [Neural ODE Control for Trajectory Approximation of Continuity Equation](https://mlsanity.com/paper/2205.09241/) (2022)
- [Deep Neural Network Approximation of Invariant Functions through Dynamical Systems](https://mlsanity.com/paper/2208.08707/) (2022)
- [Control of neural transport for normalizing flows](https://mlsanity.com/paper/2307.07817/) (2023)
- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems](https://mlsanity.com/paper/2110.12393/) (2021)
- [Feedback Favors the Generalization of Neural ODEs](https://mlsanity.com/paper/2410.10253/) (2024)
- [Time Dependence in Non-Autonomous Neural ODEs](https://mlsanity.com/paper/2005.01906/) (2020)
- [Interpolation and approximation via Momentum ResNets and Neural ODEs](https://mlsanity.com/paper/2110.08761/) (2021)

## Abstract (arXiv)

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 (SA-NODEs). 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.