Optimal Transport and PDE - Gradient Flows in the Wasserstein Metric

Explore the intersection of optimal transport theory and partial differential equations in this 59-minute lecture by Katy Craig from UC Santa Barbara. Delve into the concept of gradient flows in the Wasserstein metric, examining topics such as the continuity equation, PDE properties, and order of convergence. Investigate the aggregation equation and its dynamics, understanding the importance of PDEs in this context. Learn about grading flow and its application to two-layer neural networks, as well as the chi-squared divergence. Gain insights into the existence and uniqueness of solutions, and develop an intuitive understanding of these complex mathematical concepts.

Introduction
Motivation
Continuity Equation
PDE Properties
Order of Convergence
Aggregation Equation
Dynamics
Why PDE
Grading flow
Twolayer neural networks
Chisquared divergence
The plan
What is perpendicular mean
When do solutions exist
Uniqueness
Intuition
Existence