The Wide and Wonderful World of Optimal Transport Theory in Physics

Abstract

Optimal transport (OT) theory, first conceived to solve problems of moving dirt, has since evolved into a powerful mathematical framework with far-reaching applications across machine learning, probability and statistics, and theoretical physics. In particle physics, OT underpins many modern machine learning algorithms and data-analysis methods. It also offers rigorous numerical strategies for non-perturbative field theory calculations and provides a framework for strengthening the connection between field theories and neural networks. In this talk, I will motivate how OT provides a unifying language for connecting ensembles of neural networks, statistical field theories, and exact renormalization group flows. I will highlight how viewing field theories as probability distributions (over an infinite dimensional function space) makes OT a natural tool, offering both numerical strategies and analytic insights. Additionally, OT and the renormalization group can be used to better understand feature learning and training dynamics in ensembles of asymptotically wide neural networks. The goal is not only to showcase the wide and wonderful reach of OT in physics, but also to illustrate how it opens new perspectives for tackling non-perturbative problems and for deepening the dialogue between physics and machine learning.