This open access book offers a comprehensive introduction to a wide variety of topics in the mathematical theory of deep learning. These include questions pertaining to the prowess of deep neural networks, the surprising effectiveness of current learning algorithms, and the puzzling capability of huge neural networks to make accurate predictions on unseen data. The text focuses on rigorous but accessible results, and provides deep insights into the reasons why this field has been so successful. The book proves useful for mathematicians with a working knowledge of probability theory, linear algebra, and analysis.

Jakob Zech is a Professor for the Mathematical Foundations of Machine Learning at Heidelberg University. He received his PhD in Mathematics at ETH Zurich in 2018. Supported by an Early Postdoc.Mobility fellowship of the Swiss National Science Foundation, he then spent a year as a postdoc at the Massachusetts Institute of Technology. In 2020 he joined Heidelberg University, where he was appointed to his current professorship in 2025. His research revolves around high-dimensional approximation, operator learning, measure transport and particle-based sampling, Bayesian inference and uncertainty quantification, and the theory of deep learning.

Philipp Petersen is an Associate Professor for Mathematics of Machine Learning at the University of Vienna. He received his PhD at the TU Berlin in 2016 in the field of applied harmonic analysis. After the PhD, he carried out a Post-Doc at the University of Oxford in the framework of a DFG research scholarship. In 2019, he started a tenure-track assistant professorship in Vienna. His research revolves around deep neural networks, applied harmonic analysis, approximation theory, and numerical analysis of partial differential equations.