Courses

Mathematical Data Science and Machine Learning (Master of Mathematical Data Science)

Lecturer: Prof. Leon Bungert

4 h Tue 10-12, SE 30 and Thu 10-12, 00.102 (Library & Seminar Centre)

Exercise to Mathematical Data Science and Machine Learning (Master)

2 h Thu 16-18, 00.102 (Library & Seminar Centre)

Discrete Mathematics: PDEs on Graphs (Master)

Lecturer: Dr. Eloi Martinet

This course will be devoted to the mathematical study of famous unsupervised and semi-supervised learning algorithms like Spectral Clustering and Google's PageRank. We will see that in the large data limit, these algorithms approximate certain PDEs

3 h Mon 14-16 01.101 and Wed 10-11, 00.101 (Library & Seminar Centre)

Exercise to Discrete Mathematics: PDEs on Graphs (Master)

1 h Wed 11-12, 00.101 (Library & Seminar Centre)

Bachelor and Master seminar: Mathematics of Machine Learning

Lecturer:  Prof. Dr. Leon Bungert, Dr Eloi Martinet and Yannick Lunk

2 h, Mon 10–12, 01.104 (Library & Seminar Centre)

The theme is ‘Regularisation methods in machine learning’ and topics will range from classical regularisation methods for linear systems and inverse problems in imaging, through implicit regularisation of optimisation algorithms, to modern machine learning methods such as dropout and adversarial training.

The seminar is open to both Bachelor’s and Master’s students in mathematics, and topics will be allocated accordingly.

The first compulsory session is on Monday, 12 October 2026, 10.00–12.00, in S1.104.

We hope to welcome many of you there. Depending on the number of attendees, the actual talks will be held in a block format towards the end of the semester.

Statistical analysis of graph-based learning on manifolds (Giovanni Prodi Lecture, 4+2)

Lecturer: Prof. Dr Nicolás García Trillos

Abstract: This course focuses on the statistical understanding of the use of graphs to solve certain learning tasks. The fundamental high-level questions we will be investigating are as follows: In what sense are graph-based learning procedures statistically optimal for the underlying estimation problems they seek to address? Are there any alternative ways of processing data that could outperform existing graph-based methodologies? To answer these questions theoretically, we will explore some of the existing frameworks for studying the optimality of estimators in a general statistical setting. We will focus on the concepts of minimaxity and (classical) asymptotic efficiency of estimators, but towards the end of the course we will also discuss a notion of efficiency in terms of sensitivity to data perturbations. We will use these frameworks to explore graph-based learning on manifolds (i.e., a setting where data points are sampled from a manifold) from a well-defined statistical perspective and provide some answers to the high-level questions posed earlier. Whilst the emphasis will be on analysing graph Laplacians and their eigenpairs (in supervised and unsupervised settings), much of the discussion throughout the course is intended to spark new ideas and motivate new problems that combine PDEs, PDEs on graphs, and statistical analysis.

The course combines statistical theory and mathematical analysis. Familiarity with some probability theory, particularly concentration inequalities, as covered, for example, in the courses ‘Mathematical Data Science and Machine Learning’ or ‘PDEs on Graphs’, is required.

with practical sessions

Machine Learning and Numerics Lab (Bachelor’s in Mathematical Data Science)

Lecturer: Dr Sebastian Scott

Seminar: Finite Element Methods and Physics-Informed Neural Networks

Lecturer: Dr Eloi Martinet

Mathematical Data Science and Machine Learning (Master Mathematical Data Science)

Lecturer: Prof. Leon Bungert

with exercise

Discrete Mathematics: PDEs on Graphs (Master)

Lecturer: Dr Eloi Martinet

with exercise

Theory and Applications in Learning:

This course will be devoted to the mathematical study of famous unsupervised and semi-supervised learning algorithms like Spectral Clustering and Google's PageRank. We will see that in the large data limit, these algorithms approximate certain PDEs

Mathematical Foundations of Data Science II (Bachelor)

Lecturer: Prof. Leon Bungert

The lecture “Mathematical Foundations of Data Science” deals with the essential mathematical concepts that are essential for understanding and applying data science and machine learning. For this purpose, there are introductions to basic mathematical concepts of linear algebra and statistics. Specific mathematical techniques and methods used in data analysis and machine learning are also introduced. These include optimization, numerical methods, linear regression, cluster analysis, dimensionality reduction, artificial neural networks and deep learning. The lecture focuses on teaching students the basic methods and concepts of data science and how to use them for applications.

with Exercise

Seminar Machine Learning (Bachelor seminar)

Lecturer: Prof. Dr. Leon Bungert und Dr. Eloi Martinet

Mathematical Foundations of Data Science (Mathematik 3 for KIDS)

(2+1 im Bachelor Mathematical Data Science)­­

Lecturer: Dr. Eloi Martinet

The lecture “Mathematical Foundations of Data Science” deals with the essential mathematical concepts that are essential for understanding and applying data science and machine learning. For this purpose, there are introductions to basic mathematical concepts of linear algebra and statistics. Specific mathematical techniques and methods used in data analysis and machine learning are also introduced. These include optimization, numerical methods, linear regression, cluster analysis, dimensionality reduction, artificial neural networks and deep learning. The lecture focuses on teaching students the basic methods and concepts of data science and how to use them for applications.

with Exercise

Discreet Mathematics (PDEs on Graphs: Theory and Applications in Learning) (Master)

Lecturer: Prof. Dr. Leon Bungert

with Exercise

Machine Learning with Graphs (Master Seminar)

Lecturer: Prof. Dr. Leon Bungert, Dr. Eloi Martinet

In this seminar we will discover machine learning methods that involve graphs. This includes partial differential equations on graphs and their use for semi-supervised machine learning, as well as graph neural networks for supervised learning with graph data. The seminar will cover theoretical and numerical aspects and can lead to a master's thesis in this topic.

Referenzen:

[1] Calder, J., Cook, B., Thorpe, M., & Slepcev, D. (2020, November). Poisson learning: Graph based semi-supervised learning at very low label rates. In International Conference on Machine Learning (pp. 1306-1316). PMLR.

[2] Calder, J. (2018). The game theoretic p-Laplacian and semi-supervised learning with few labels. Nonlinearity, 32(1), 301.

[3] Bungert, L., Calder, J., & Roith, T. (2023). Uniform convergence rates for Lipschitz learning on graphs. IMA Journal of Numerical Analysis, 43(4), 2445-2495.

[4] Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., & Vandergheynst, P. (2017). Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4), 18-42.

[5] Bronstein, M. M., Bruna, J., Cohen, T., & Veličković, P. (2021). Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint arXiv:2104.13478.

[6] Xia, F., Sun, K., Yu, S., Aziz, A., Wan, L., Pan, S., & Liu, H. (2021). Graph learning: A survey. IEEE Transactions on Artificial Intelligence, 2(2), 109-127.

[7] Song, Z., Yang, X., Xu, Z., & King, I. (2022). Graph-based semi-supervised learning: A comprehensive review. IEEE Transactions on Neural Networks and Learning Systems.­­

Mathematical Foundations of Data Science II (2+1 in the Bachelor Mathematical Data Science)

Dozent: Prof. Leon Bungert

The lecture “Mathematical Foundations of Data Science” deals with the essential mathematical concepts that are essential for understanding and applying data science and machine learning. For this purpose, there are introductions to basic mathematical concepts of linear algebra and statistics. Specific mathematical techniques and methods used in data analysis and machine learning are also introduced. These include optimization, numerical methods, linear regression, cluster analysis, dimensionality reduction, artificial neural networks and deep learning. The lecture focuses on teaching students the basic methods and concepts of data science and how to use them for applications.

with Exercise

Numerical Mathematics and Applied Analysis (Work group)

Lecturer: Dr. Eloi Martinet

In the vast majority of theoretical and real-world applications, the solution to a partial differential equation can not be computed analytically. The aim of this course is to explore two methods allowing to compute approximate solutions. The first, ”traditional” one, is the Finite Element Method. The second one makes use of the recent developments in Deep Learning leading to the so called ”Physics Informed Neural Networks”. In a first part, we will study the fundamental tools needed to solves elliptic partial differential equations, such as weak derivatives and Sobolev spaces. We will show how to approximate PDEs using basic finite elements tools. In the second part, we present the definition of au Neural Network and derive the proof of the so-called "Universal Approximation Theorem". Using backpropagation, we show how a network can be trained to solve some partial differential equations.

• Mathematical Foundations of Data Science

Lecturer: Prof. Leon Bungert