# Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems

arXiv: https://arxiv.org/abs/2110.12393 · HTML: https://mlsanity.com/paper/2110.12393/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 24 Oct 2021.

## Builds on (most cited first, 11 of 11)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Convolutional Neural Networks at Constrained Time Cost](https://mlsanity.com/paper/1412.1710/) (2014)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [A Measure Theoretical Approach to the Mean-field Maximum Principle for Training NeurODEs](https://mlsanity.com/paper/2107.08707/) (2021)
- [Control in the spaces of ensembles of points](https://mlsanity.com/paper/1907.00905/) (2019)
- [A gradient flow equation for optimal control problems with end-point cost](https://mlsanity.com/paper/2107.00556/) (2021)
- [Mean-Field Pontryagin Maximum Principle](https://mlsanity.com/paper/1504.02236/) (2015)
- [Ensemble controllability by Lie algebraic methods](https://mlsanity.com/paper/1603.07133/) (2016)
- [Optimal control on distributions](https://mlsanity.com/paper/1702.02750/) (2017)

## Built on by (most cited first, 8 of 8)

- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)
- [Autonomous-Flow-Based Generation](https://mlsanity.com/paper/2511.09902/) (2025)
- [Constructive interpolation and generalization rates for neural ODEs: a control perspective](https://mlsanity.com/paper/2606.00469/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

In this paper we propose a Deep Learning architecture to approximate diffeomorphisms diffeotopic to the identity. We consider a control system of the form dot x = sumi=1lFi(x)ui, with linear dependence in the controls, and we use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points. Despite the simplicity of the control system, it has been recently shown that a Universal Approximation Property holds. The problem of minimizing the sum of the training error and of a regularizing term induces a gradient flow in the space of admissible controls. A possible training procedure for the discrete-time neural network consists in projecting the gradient flow onto a finite-dimensional subspace of the admissible controls. An alternative approach relies on an iterative method based on Pontryagin Maximum Principle for the numerical resolution of Optimal Control problems. Here the maximization of the Hamiltonian can be carried out with an extremely low computational effort, owing to the linear dependence of the system in the control variables.