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[Theme]Machine Learning from an Applied Mathematician's Perspective

[Date & Time] March 19, 10:00

[Location]Room 603, Lide Building, Zhongguancun Campus

[Speaker]Enrique Zuazua

[Abstract]Machine Learning has emerged as one of the most transformative forces in contemporary science and technology. In this lecture, I discuss Machine Learning through the lens of applied mathematics, highlighting its connections with control theory, partial differential equations, and numerical analysis. The presentation is organized around three goals: representation, generalization, and generation.

We revisit the links between Machine Learning and systems control (cybernetics),interpreting representation and expressivity in deep neural networks in terms of ensemble controllability of neural differential equations. Within this framework, generalization appears as a stability property with respect to perturbations in the data and the model.Next, we discuss neural-network architectures as tools for numerical approximation, using the Dirichlet problem for the Laplace equation formulated as an energy minimization problem under neural-network constraints. Finally, we present a PDE-based perspective on generative diffusion models, interpreting their convergence through the asymptotic behavior of Fokker-Planck equations driven by the score field, and highlighting how classical tools shed light on their regularization and convergence properties.