The GdT OT-PDE-ML is a research seminar organized by the ParMA team. It brings together researchers to discuss recent theoretical and numerical developments at the crossroads of Optimal Transport theory, Partial Differential Equations and Machine Learning.

Abstract: Variational mean-field games are a class of mean-field games whose equilibria can be characterized as minimizers of a global energy functional combining agents’ individual costs and collective interactions. In the classical formulation, the energy functional resembles the (Eulerian) Benamou-Brenier type formula for entropic optimal transport, with the addition of a convex term which encompasses the interaction between the agents. In analogy with entropic OT, Benamou, Carlier, Di Marino and Nenna recently proposed an equivalent (Lagrangian) entropy-minimization formulation of the problem. The aim of this talk is two-fold. First, we investigate the small-noise Gamma limit for the lagrangian entropy-minimization formulation, establishing Gamma convergence of noisy second order mean-field games towards its deterministic first order counterpart. This also translates into a dynamical proof of the convergence of entropic OT to classical OT. Secondly, we study the original Eulerian problem. We prove a first order energy expansion, which establishes a selection principle for minimizers of the deterministic problem. This is a joint work with Luca Nenna.