Course Title: PDEs Meet Machine Learning: Integrating Numerics, Control, and Machine Learning
Lecturer: Professor Enrique Zuazua (Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany)
Course Dates: July 21–23, 2025, 9:00 – 11:00
Venue: Room 209, Zhengxin Building, Jilin University, Changchun; Zoom ID: 904 645 6677, Passcode: 2025
Abstract:
Partial Differential Equations (PDEs) form the cornerstone of mathematical modeling in mechanics and the natural sciences, driving advances in analysis, numerical methods, and applied mathematics. Today, the rise of Machine Learning (ML) and Artificial Intelligence (AI) presents transformative opportunities and challenges for classical PDE methodologies. Can ML enhance PDE techniques without sacrificing mathematical rigor? Can we develop hybrid computational frameworks that leverage data-driven approaches while maintaining the reliability of traditional methods?
This lecture explores these questions through an interdisciplinary lens, bridging PDE theory, control, and ML. We examine the intrinsic connections between representation, optimization, and control theory—rooted in cybernetics (from Ampère to Wiener) and historically motivated by the quest to design intelligent machines. Interestingly, the goals of control theory align closely with those of modern AI, emphasizing mathematics’ unifying power in modeling and innovation.
We discuss recent work addressing two key challenges: Why does ML generalize so effectively? and How can data-driven insights be rigorously integrated into classical applied mathematics, particularly for PDEs and numerical methods? This exploration is shaping a new paradigm of PDE+D(ata), to forge the next generation of computational tools.
Lecturer Biography:
Enrique Zuazua is a Professor of Mathematics at Friedrich-Alexander-Universität Erlangen-Nürnberg, a Member of the European Academy of Sciences, and an internationally leading expert in control theory for distributed parameter systems. He received the W.T. and Idalia Reid Prize from the Society for Industrial and Applied Mathematics (SIAM). He has served as an editorial board member, associate editor, or editor-in-chief for journals including Journal de Mathématiques Pures et Appliquées, SIAM Journal on Control and Optimization, Journal of Differential Equations, and ESAIM: Control, Optimisation and Calculus of Variations. He delivered a 45-minute invited lecture at the 2006 International Congress of Mathematicians and has been invited to speak at the 2026 International Congress of Mathematicians.
