Event: MFML 2026, Mathematical Foundations of Machine Learning
Date: Thu. May 25 – 29, 2026

Lecture: Machine Learning from an Applied Mathematician’s Perspective
Speaker: Prof. Enrique Zuazua, FAU – Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

Abstract. Machine Learning has emerged as one of the most transformative forces in contemporary science and technology. In this three-lecture series, I will discuss Machine Learning through the lens of applied mathematics, emphasizing its connections with control theory, partial differential equations, and numerical analysis.

In the first lecture, we will revisit the historical and conceptual links between Machine Learning and Systems Control (Cybernetics). This point of view allows us to reinterpret representation and expressivity properties of deep neural networks in terms of ensemble or simultaneous controllability of neural differential equations.

The second lecture will focus on the use of neural network architectures as numerical approximation tools. We will consider, as a guiding example, the classical Dirichlet problem for the Laplace equation, formulated via energy minimization under neural-network constraints. Particular attention will be paid to the lack of convexity and coercivity in the resulting optimization problems. We will show how relaxation techniques may restore convexity at the price of losing coercivity, and we will discuss the mathematical implications of this trade-off for analysis and computation.

In the third lecture, we will present a PDE-based perspective on generative diffusion models. Their convergence can be reinterpreted in terms of the asymptotic behavior of Fokker–Planck equations driven by the so-called score vector field. We will explain how classical tools, such as Li-Yau-type differential inequalities for positive solutions of the heat equation, provide insight into the regularization and convergence properties of these models.

The series will conclude with a discussion of open problems and promising directions for future research at the interface of control theory, PDEs, numerical analysis, and modern Machine Learning.

Lecturers
• Lénaïc Chizat. EPFL
• Nicolás García-Trillos. University of Wisconsin-Madison
• Guido Montúfar. UCLA | MPI-Leipzig
• Enrique Zuazua. FAU

WHEN
Thu. May 28, 2026 at HH:00H (local time)

WHERE
On-site. The School will be hosted at:
Hotel Montana. Vason – Trento (Italy)

Organizers
Gian Paolo Leonardi. Department of Mathematics, University of Trento
Giuseppe Savaré. Department of Decision Sciences, Bocconi University, Milan

See more at the official page of the event