Nicolas Masson et Borjan Geshkovski

Nicolas Masson (Université Paris-Saclay)

 

Title : Modeling « polite » crowds : asymptotics of weighted projection problems 

 

Abstract : In 2010, Aude-Roudneff Chupin proposed in her PhD thesis a new macroscopic crowd motion model, yielding an unclassified PDE that couples a continuity equation and a Hilbertian projection problem. Together with Bertrand Maury and Filippo Santambrogio, they proved the existence of a solution to this new PDE system, and enhanced the underlying gradient flow structure using a JKO scheme. However, this model allowed individuals to travel faster than their desired velocity - which contradicts several empirical laws in crowd motion. The aim of this talk will be to present related optimization problems that were introduced to correct these modeling issues, and show how the study of these problems helps address the challenge of modeling « polite » crowds.

 

 

Borjan Geshkovski (LJLL, INRIA)

Title: Approximate conditional flow matching

 

Abstract: In the context of (normalizing) flow matching, one parametrizes the vector field of a continuity equation using a two-layer neural network and fits the parameters to minimize a discrepancy between the resulting solution and an unknown target measure. We focus on the conditional transport problem, where the goal is also to approximate the transport map that pushes forward the initial condition to the unknown target measure. We provide an explicit construction of parameters that are piecewise constant in time, enabling the simultaneous approximation of both the measure (in total variation) via the continuity equation and the transport map (in L2) via the associated solution map. This construction has the desirable property that the resulting solution map closely resembles the Knothe–Rosenblatt rearrangement between suitable discretizations of the measures.