Diffusion Models: Theory and Applications (in PDEs)

Applied and Computational Mathematics Seminar
Monday, April 8, 2024 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005 and https://gatech.zoom.us/j/98355006347
Yulong Lu – University of Minnesota, Twin Cities – yulonglu@umn.eduhttps://lu.math.umn.edu/
Wenjing Liao

Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis show that SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by certain low-complexity structures.  In the second part of the talk, we consider the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present the development of a physics-guided diffusion model designed for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation.