Abstract: Diffusion models and flow matching have achieved remarkable progress in image generation. This talk emphasizes their interpretation as dynamic partial differential equations that transport the data distribution to a known prior via a learned velocity field. That viewpoint exposes their theoretical underpinnings and grounds our discussion of some key issues and current progress. (1) Scientific data often lies on a lower-dimensional constraint set, where no ambient density exists and both scores and Jacobian log-determinants blow up; I present a constraint-aware perturbation that restores stable training and then show where its guarantees stop. (2) The velocity field's regularity at the endpoints, (3) the gap between an objective whose exact minimizer memorizes and models that nonetheless generalize, and (4) the relationship between straight trajectories and optimal ones have all seen recent answers. The talk will illustrate those results and some open issues with numerical examples.
Bio: Lars Ruthotto is a Professor in the Departments of Mathematics and Computer Science at Emory University, where he is a member of the Scientific Computing Group and directs the Emory REU/RET Site for Computational Mathematics for Data Science. He earned his PhD in Mathematics from the University of Muenster in 2012 and was a postdoc at the University of British Columbia before joining Emory. His research develops computational methods at the interface of scientific computing, optimization, machine learning, and inverse problems, with recent emphasis on generative modeling, high-dimensional partial differential equations, and stochastic optimal control. He is a recipient of the NSF CAREER Award and serves as a Section Editor for the SIAM Journal on Scientific Computing.