Mathematical Foundations of Control and Machine Learning

Friedrich-Alexander-Universität Erlangen-Nürnberg — Department of Mathematics · Master’s course, Winter 2026

Lecturer	Prof. Dr. Enrique Zuazua
Teaching Assistant	Daniel López Montero
Location	12 October: C3 – Chemikum (Nikolaus-Fiebiger-Straße 10, 91058 Erlangen) All other sessions: H21 (Cauerstraße 5b, 91058 Erlangen)
Evaluation	Oral presentation (100%); at least 80% attendance is required to take part in the final evaluation
Contact	cml.fau@gmail.com
StudOn	studon.fau.de/campo/course/579004

About the course

The course provides a unified mathematical perspective on control theory and machine learning. It introduces key concepts from the analysis of dynamical systems governed by ordinary and partial differential equations (ODEs and PDEs) — controllability, observability and stability — and builds on them to show how control-theoretic and optimization-based methods explain the dynamics of state-of-the-art machine learning systems: deep neural networks, federated learning frameworks, reinforcement learning and large language models (LLMs).

Learning objectives

By the end of the course students will have developed a solid understanding of the mathematical and dynamical foundations underlying control and learning systems. They will be able to

• analyse the optimization and stability properties of deep learning algorithms,
• explain how control-theoretic principles extend to large-scale models such as LLMs,
• deliver clear and rigorous scientific presentations of technical concepts and research results.

Evaluation

Assessment takes into account attendance and active participation and a group presentation in the final sessions of the course. Attendance of at least 80% of the lectures is required to take part in the final evaluation.

• Groups of at least three students.
• Content: any topic related to the course, provided it has substantial mathematical content. Examples of presentations from previous years: 1, 2 and 3; see also the list of topics presented in previous years.
• Registration: before 1 November 2026, email cml.fau@gmail.com the names of all group members, the topic and your preferred presentation dates.
• Slides (PDF) are sent to the same address before the day of the presentation.
• Format: 15 minutes of presentation and 15 minutes of questions; every member takes part in both.
• Attendance at all presentation sessions is required, also after your own group has presented.

Lectures	12 Oct – 7 Dec 2026
Presentations	14 & 21 Dec 2026 · 11, 18, 25 Jan & 1 Feb 2027 (schedule)

Prerequisites

Strongly recommended: basic knowledge of calculus, linear algebra, ODEs and PDEs. The code is written in Python with PyTorch, but no prior exposure to either is assumed — the notebooks keep the code to the shortest fragment that illustrates the mathematics.

Course material

Each notebook is self-contained: the mathematics is developed in the text and the code is the smallest runnable confirmation of it. Code cells are folded; click Code to expand them.

Lecture slides

#	Lecture	Slides	Topics
00	Introduction to Control and Machine Learning	PDF · PowerPoint	The context today: AI everywhere, representation, the risk of hallucinations; origins: Aristotle, Watt and Maxwell, Wiener and Kalman, Markov and PageRank, herding and parking; the objectives of machine learning, universal approximation, shallow and deep networks, supervised learning, least squares; the 3Blue1Brown videos
01	Introduction to Control Theory	PDF	A short history; dynamical systems; the ingredients of a control problem; optimal control, controllability and feedback; reachable sets, the Kalman rank condition, the Gramian and the cost of control; control and learning

Part 0 — The map

#	Notebook	Topics
00	The Big Picture: Control and Machine Learning	The three control problems and the learning problem, the course map, a control ↔︎ ML dictionary, the recurring examples

Part I — Control

#	Notebook	Topics
01	What is Control?	States, controls, targets; optimal control vs. controllability vs. stabilization; open and closed loop; the overdamping paradox; a short history
02	Linear Control Systems and the Kalman Rank Condition	Variation of constants, reachable sets, the Kalman theorem with proof, the controllability Gramian and an explicit steering control, quantitative controllability
03	Observability, Duality and the Construction of Controls	The adjoint system, the duality identity, the HUM functional, observability and coercivity, minimal-norm controls, conditioning and gradient descent
04	Bang-Bang and Switching Controls	Structured controls from modified dual functionals: minimal-amplitude bang-bang controls, switching controls, Hölder duality
05	Feedback Stabilization, LQR and MPC	Stabilization from observability, the overdamping ceiling, pole placement, LQR and the Riccati equation, model predictive control

Part II — Optimization and long-time control

#	Notebook	Topics
06	Minimizers, Gradient Flows and Gradient Descent	Lagrange multipliers, existence of minimizers, the gradient flow and its Lyapunov function, step sizes and conditioning, Snell’s law, optimal transport
07	Adjoint Methods, Density and Control of the Heat Equation	The adjoint method (two solves per gradient), density and approximate controllability, null controllability of the heat equation, where observability fails
08	Long-Time Optimal Control and the Turnpike Property	The steady optimum, forward–backward optimality systems, hyperbolicity of the Hamiltonian matrix, the exponential turnpike theorem

Part III — Machine learning

#	Notebook	Topics
09	Machine Learning Foundations: Fitting Is Not Learning	Risks and representations, models and training, backpropagation, overfitting and selection bias, the train/validation/test protocol, federated learning, a zoo of architectures
10	Training from Data: Gradient Descent, SGD and Mini-Batches	The descent lemma, convergence guarantees, stochastic gradients and their noise, early stopping, Langevin dynamics and the Gibbs measure
11	Optimization Algorithms: Momentum and Adaptive Methods	Heavy ball and Nesterov momentum as damped dynamics, RMSProp and Adam, fair comparisons of optimizers

Part IV — Neural networks as dynamical systems

#	Notebook	Topics
12	Shallow Networks: Representation, Sparsity, Convexity	Universal approximation, the outer-weight convexification, sparsity via linear programming
13	ResNets and Neural ODEs	Residual networks as Euler schemes, numerical ODE solvers, the error of Euler’s method, training as simultaneous optimal control, backpropagation as the discrete adjoint, learning the Lorenz dynamics, flows that deform the plane
14	Classification as Simultaneous Control	The classification theorem, the three exact actions of a ReLU neuron, explicit switching controls, switch counts and width vs. depth, a pre-registered comparison of constructed and trained classifiers
15	Transformers: Attention and Token Dynamics	Self-attention, multi-head attention, positional encoding, a character-level BERT on Tiny Shakespeare, hardmax attention as interacting particles, the clustering theorem
16	Neural Transport and Learning Dynamics from Data	Push-forward and the transport equation, the Wasserstein-1 distance, atoms versus laws, semi-autonomous neural ODEs, Gronwall estimates
17	Diffusion Models	The heat equation as noising, Feynman–Kac, Ornstein–Uhlenbeck forward process, reverse-time SDE and probability-flow ODE, score matching

Part V — Closing the loop

#	Notebook	Topics
18	Reinforcement Learning	Bellman equation and value iteration, Q-learning, exploration vs. exploitation, Pontryagin’s maximum principle, Hamilton–Jacobi–Bellman (appendix)
19	Control and Learning: A Synthesis	The three threads of the course, the dictionary with its scope, one oscillator hosting four different problems, capacity vs. optimization vs. generalization

Installation and running the notebooks

The source is on GitHub. The python environment is managed with uv.

git clone https://github.com/cml-fau/cml-fau.github.io.git mfcml
cd mfcml
uv sync # This creates a `.venv/` with all python dependencies installed

To run the notebooks, open one in VS Code and select .venv/bin/python as the kernel.

Without uv

python3 -m venv .venv
.venv/bin/pip install --index-url https://download.pytorch.org/whl/cpu torch
.venv/bin/pip install numpy matplotlib jupyter ipykernel

In JupyterLab

uv run jupyter lab notebooks/

Bibliography

Video material

The 3Blue1Brown neural networks series is the standard visual companion to notebooks 09–11 and 15:

1. But what is a neural network?
2. Gradient descent, how neural networks learn
3. What is backpropagation really doing?
4. Backpropagation calculus

The newer chapters of the same series cover large language models and pair well with notebook 15: But what is a GPT? and Attention in transformers, visually explained.

Books and articles

1. I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning. MIT Press, 2016.
2. R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction, 2nd ed. MIT Press, 2018.
3. C. F. Higham and D. J. Higham, “Deep learning: An introduction for applied mathematicians,” SIAM Review, 61(4) (2019), 860–891.
4. L. Bottou, F. E. Curtis, and J. Nocedal, “Optimization methods for large-scale machine learning,” SIAM Review, 60(2) (2018), 223–311.
5. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. Springer, 2006.
6. J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136. American Mathematical Society, 2007.
7. D. Ruiz-Balet and E. Zuazua, “Neural ODE control for classification, approximation and transport,” arXiv preprint arXiv:2104.05278, 2021.
8. E. Zuazua, “Propagation, observation, and control of waves approximated by finite difference methods,” SIAM Review, 47(2) (2005), 197–243.
9. E. Zuazua, “Controllability and observability of partial differential equations: Some results and open problems,” in Handbook of Differential Equations: Evolutionary Equations, Vol. 3, North-Holland, 2006, pp. 527–621.