# Machine Learning and Control: Foundations, Advances, and Perspectives

arXiv: https://arxiv.org/abs/2510.03303 · HTML: https://mlsanity.com/paper/2510.03303/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Understanding and Improving Transformer From a Multi-Particle Dynamic System Point of View](https://mlsanity.com/paper/1906.02762/) (2019)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Towards Understanding the Condensation of Neural Networks at Initial Training](https://mlsanity.com/paper/2105.11686/) (2021)
- [Clustering in pure-attention hardmax transformers and its role in sentiment analysis](https://mlsanity.com/paper/2407.01602/) (2024)
- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)
- [Control of neural transport for normalizing flows](https://mlsanity.com/paper/2307.07817/) (2023)
- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Approximate and Weighted Data Reconstruction Attack in Federated Learning](https://mlsanity.com/paper/2308.06822/) (2023)
- [Cluster-based classification with neural ODEs via control](https://mlsanity.com/paper/2312.13807/) (2023)
- [FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning](https://mlsanity.com/paper/2402.13989/) (2024)
- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [Stability and Convergence of a Randomized Model Predictive Control Strategy](https://mlsanity.com/paper/2211.05463/) (2022)
- [A framework for randomized time-splitting in linear-quadratic optimal control](https://mlsanity.com/paper/2109.13798/) (2021)
- [Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics](https://mlsanity.com/paper/2412.01619/) (2024)
- [HYCO: Hybrid-Cooperative Learning for Data-Driven PDE Modeling](https://mlsanity.com/paper/2509.14123/) (2025)
- [Model predictive control with random batch methods for a guiding problem](https://mlsanity.com/paper/2004.14834/) (2020)
- [Moments, Time-Inversion and Source Identification for the Heat Equation](https://mlsanity.com/paper/2507.02677/) (2025)

## Abstract (arXiv)

Control theory of dynamical systems offers a powerful framework for tackling challenges in deep neural networks and other machine learning architectures. We show that concepts such as simultaneous and ensemble controllability offer new insights into the classification and representation properties of deep neural networks, while the control and optimization of static systems can be employed to better understand the performance of shallow networks. Inspired by the classical concept of turnpike, we also explore the relationship between dynamic and static neural networks, where depth is traded for width, and the role of transformers as mechanisms for accelerating classical neural network tasks. We also exploit the expressive power of neural networks (exemplified, for instance, by the Universal Approximation Theorem) to develop a novel hybrid modeling methodology, the Hybrid-Cooperative Learning (HYCO), combining mechanics and data-driven methods in a game-theoretic setting. Finally, we describe how classical properties of diffusion processes, long established in the context of partial differential equations, contribute to explaining the success of modern generative artificial intelligence (AI). We present an overview of our recent results in these areas, illustrating how control, machine learning, numerical analysis, and partial differential equations come together to motivate a fertile ground for future research.