Mathematics 690STF - Mathematics/GenerativeModeling

In the last decade, generative models and generative artificial intelligence have produced breakthrough results in image generation, text and speech synthesis, and more recently in scientific research itself, in fields such as aerospace, astronomy, biology, materials science, and medicine. Not unexpectedly, mathematics plays a foundational role in generative AI. It is essential for gaining a deeper understanding of existing methods, establishing limitations, quantifying trustworthiness, and developing new, provably more robust, or more energy-efficient methods. Specifically, we highlight the pivotal role played by dynamical systems, stochastic differential equations, and partial differential equations in the development of some of the most recent and successful generative models. In this course, we will cover two main aspects of the field. (1) Mathematical Foundations for Generative Modeling: This module will provide a unified mathematical background presented in the context of generative modeling, that includes elements of applied probability, statistical inference, optimal transport, information theory and optimization methods. (2) Current methods, algorithms, and significant challenges in generative modeling: We will study various state-of-the-art generative modeling techniques, including Normalizing Flows and Neural Differential Equations, Generative Adversarial Networks (GAN), Variational Autoencoders (VAE), Energy-based Models, Deep Autoregressive Models & Bayesian Networks, Diffusion and score-based Models, and Flow-matching methods. Lastly, we will discuss ongoing challenges in developing energy-efficient methods (distillation and one-shot methods) and recently identified issues with the memorization of data.

This course is only for graduate students. Students should have a basic understanding of probability, differential equations, linear algebra, and computation. For instance, at least at the level of Stat 607; Math 651; Math 534 and/or Math 53 & Math 545.