# Expressivity of Quadratic Neural ODEs

arXiv: https://arxiv.org/abs/2504.09385 · HTML: https://mlsanity.com/paper/2504.09385/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 17 references that are in the map.
- First posted on arXiv 13 Apr 2025.

## Builds on (most cited first, 4 of 4)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [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)
- [Approximation capability of two hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/2101.09181/) (2021)

## Built on by (most cited first, 1 of 1)

- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

This work focuses on deriving quantitative approximation error bounds for neural ordinary differential equations having at most quadratic nonlinearities in the dynamics. The simple dynamics of this model form demonstrates how expressivity can be derived primarily from iteratively composing many basic elementary operations, versus from the complexity of those elementary operations themselves. Like the analog differential analyzer and universal polynomial DAEs, the expressivity is derived instead primarily from the "depth" of the model. These results contribute to our understanding of what depth specifically imparts to the capabilities of deep learning architectures.