# On the space-time expressivity of ResNets

arXiv: https://arxiv.org/abs/1910.09599 · HTML: https://mlsanity.com/paper/1910.09599/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2019.

## Built on by (most cited first, 3 of 3)

- [Universal Approximation of Residual Flows in Maximum Mean Discrepancy](https://mlsanity.com/paper/2103.05793/) (2021)
- [Approximation properties of Residual Neural Networks for Kolmogorov PDEs](https://mlsanity.com/paper/2111.00215/) (2021)
- [ADAPT: Lightweight, Long-Range Machine Learning Force Fields Without Graphs](https://mlsanity.com/paper/2509.24115/) (2025)

## Abstract (arXiv)

Residual networks (ResNets) are a deep learning architecture that substantially improved the state of the art performance in certain supervised learning tasks. Since then, they have received continuously growing attention. ResNets have a recursive structure xk+1 = xk + Rk(xk) where Rk is a neural network called a residual block. This structure can be seen as the Euler discretisation of an associated ordinary differential equation (ODE) which is called a neural ODE. Recently, ResNets were proposed as the space-time approximation of ODEs which are not of this neural type. To elaborate this connection we show that by increasing the number of residual blocks as well as their expressivity the solution of an arbitrary ODE can be approximated in space and time simultaneously by deep ReLU ResNets. Further, we derive estimates on the complexity of the residual blocks required to obtain a prescribed accuracy under certain regularity assumptions.