Generative AI has advanced rapidly, with breakthrough results ranging from image, speech and text analysis to scientific research itself. These advances rest in part on massive computational power, large datasets, large language models, and specialized architectures that draw from mathematical concepts and techniques ranging from high-dimensional linear algebra, probability and dynamical systems to scaling limits, phase transitions and information theory. What is still missing is a deep conceptional understanding of the underlying learning and optimization mechanisms and of the complex emergent phenomena they exhibit. Further progress will depend not only on larger models and greater computational resources, but on deeper theoretical understanding, from which new mathematical questions are emerging. Such understanding has direct practical consequences: it is what can make learning methods trustworthy and certifiable, and what can yield lightweight, energy-efficient algorithms able to operate on the potentially scarce but highly structured data characteristic of scientific AI.

In this spirit we organize a week-long working group bringing together scientists from different areas, contributing distinct viewpoints while sharing a common rigorous mathematical language. Within the general framework of continuum mechanics modelling, of non-equilibrium statistical mechanics techniques and of numerical analysis methodology, the themes we intend to investigate include scaling limits of diffusive models, gradient flows and measure transport; mean-field limits, control and games; information-theoretic principles for building large language model architectures; operator learning and foundation models for PDEs; uncertainty quantification, error control and performance guarantees; statistical and Bayesian approaches to inverse problems; and graph theory and neural networks.

The structure is deliberately flexible. The first two days are devoted to introductory presentations motivating the free discussion sessions that follow, with additional short talks arranged as topics arise. Everybody interested is cordially invited to participate. Please drop us a message and we keep you updated.

Invited speakers and schedule (25-29 January 2027):

Monday — Markos Katsoulakis, Luc Rey-Bellet, Benjamin Zhang

Tuesday — Davide Gabrielli, Filippo Mignosi, Charalambos Makridakis, Masoumeh Dashti

Wednesday-Friday: free discussion sessions and short talks

Organizers:

Diego Alberici

Davide Gabrielli

Tobias Kuna

Filippo Mignosi

Dimitrios Tsagkarogiannis