Probability flows and stochastic processes: Applications to optimal transport and generative modeling

This project involves the study of probability flows to enhance understanding of complicated probability distributions, with applications ranging from the distribution of a cloud of electrons in a given molecule as building blocks of materials science, to the distribution of all possible coherent sentences in the English language which lies at the core of the advancements of large language models in artificial intelligence. In one direction, the project will propose a novel framework that facilitates the usage of probability flows to advance the theory and computation of multi-marginal optimal transport---a fundamental problem in fields ranging from quantum chemistry and economics to data science. In another direction, the project will investigate the structure of probability flows in broader settings, focusing in particular on lower-dimensional representations of the flows, in order to understand them better and increase their efficiency. The project will also contribute to workforce development by engaging graduate students and postdoctoral researchers in cutting-edge research and mentoring, while making all resulting software, publications, and teaching materials freely and publicly available. This project will center around two interrelated research directions. The first will develop a novel dynamical formulation of multi-marginal optimal transport and related optimization problems, such as versions of multi-marginal Schrödinger bridges. These formulations open the door to the applications of tools from convex optimization (e.g., proximal splitting methods), as well as methods from generative modeling such as flow matching, to finding quasi-Monge solutions of the multi-marginal optimal transport problem. The second direction will focus on the fundamentals of flow-based methods in generative modeling. In particular, the project will investigate the intrinsic dimensionality of probability flows via the new concept of the entropy matrix, which simultaneously generalizes the Fisher information matrix, as well as a matrix associated with optimal transport maps. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.