# Rademacher Complexity of Neural ODEs via Chen-Fliess Series

arXiv: https://arxiv.org/abs/2401.16655 · HTML: https://mlsanity.com/paper/2401.16655/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 30 Jan 2024.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth](https://mlsanity.com/paper/2003.05508/) (2020)
- [Continuous-in-Depth Neural Networks](https://mlsanity.com/paper/2008.02389/) (2020)
- [Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristics](https://mlsanity.com/paper/2105.03918/) (2021)
- [Framing RNN as a kernel method: A neural ODE approach](https://mlsanity.com/paper/2106.01202/) (2021)
- [Infinitely deep neural networks as diffusion processes](https://mlsanity.com/paper/1905.11065/) (2019)
- [Generalization bounds for neural ordinary differential equations and deep residual networks](https://mlsanity.com/paper/2305.06648/) (2023)
- [New directions in the applications of rough path theory](https://mlsanity.com/paper/2302.04586/) (2023)
- [On expansions for nonlinear systems, error estimates and convergence issues](https://mlsanity.com/paper/2012.15653/) (2020)
- [Imbedding Deep Neural Networks](https://mlsanity.com/paper/2202.00113/) (2022)

## Built on by (most cited first, 5 of 5)

- [Length independent generalization bounds for deep SSM architectures via Rademacher contraction and stability constraints](https://mlsanity.com/paper/2405.20278/) (2024)
- [On the Generalization Behavior of Deep Residual Networks From a Dynamical System Perspective](https://mlsanity.com/paper/2602.20921/) (2026)
- [A finite-sample generalization bound for stable LPV systems](https://mlsanity.com/paper/2405.10054/) (2024)
- [Dynamical Implicit Neural Representations](https://mlsanity.com/paper/2511.21787/) (2025)
- [Upper Generalization Bounds for Neural Oscillators](https://mlsanity.com/paper/2603.09742/) (2026)

## Abstract (arXiv)

We show how continuous-depth neural ODE models can be framed as single-layer, infinite-width nets using the Chen--Fliess series expansion for nonlinear ODEs. In this net, the output ``weights'' are taken from the signature of the control input -- a tool used to represent infinite-dimensional paths as a sequence of tensors -- which comprises iterated integrals of the control input over a simplex. The ``features'' are taken to be iterated Lie derivatives of the output function with respect to the vector fields in the controlled ODE model. The main result of this work applies this framework to derive compact expressions for the Rademacher complexity of ODE models that map an initial condition to a scalar output at some terminal time. The result leverages the straightforward analysis afforded by single-layer architectures. We conclude with some examples instantiating the bound for some specific systems and discuss potential follow-up work.