In this talk I will first review recent results which characterize adversarial training (AT) of binary classifiers as nonlocal perimeter regularization. Then I will speak about a probabilistic generalization of AT which also admits such a geometric interpretation, albeit with a different nonlocal perimeter. Using suitable relaxations one can prove the existence of solutions for…

We examine the asymptotic behaviors of solutions to Hamilton-Jacobi equations while varying the underlying domains. We establish a connection between the convergence of these solutions and the regularity of the additive eigenvalues in relation to the domains. To accomplish this, we introduce a framework based on Mather measures that enables us to compute the one-sided derivative…

We consider inverse problems concerning the one-dimensional viscous Burgers equation and some related nonlinear systems (involving heat effects, variable density, and fluid-solid interaction). We are dealing with inverse problems in which the goal is to find the size of the spatial interval from some appropriate boundary observations. Depending on the properties of the initial and…

I will discuss certain systems of transport type whose coefficients depend nonlinearly on the solution. Applications of such systems range from the modeling of pressure-less gases to the study of mean field games in a discrete state space. I will identify a notion of weak solution within the class of coordinate-wise decreasing functions, a condition…