# Continuous limits of residual neural networks in case of large input data

arXiv: https://arxiv.org/abs/2112.14150 · HTML: https://mlsanity.com/paper/2112.14150/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2021.

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

- [Variational formulations of ODE-Net as a mean-field optimal control problem and existence results](https://mlsanity.com/paper/2303.05924/) (2023)

## Abstract (arXiv)

Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural differential equations. For large scale input data we derive a mean--field limit and show well--posedness of the resulting description. Further, we analyze the existence of solutions to the training process by using both a controllability and an optimal control point of view. Numerical investigations based on the solution of a formal optimality system illustrate the theoretical findings.