COMPUTATIONAL MATH – IDeAS SEMINAR Recurring weekly series · Thursdays, 2:00 – 3:00 PM This week's talk: Speaker: Florian Schäfer (NYU) Title: Toward Information Geometric Mechanics Date: Thursday, April 2, 2026 Time: 2:00 PM – 3:00 PM Room: 224 Fine Hall Abstract: Shock waves in high-speed gas dynamics cause severe numerical difficulties for classical solvers and scientific machine learning. They are fundamentally a multiscale problem: While viscous effects ensure smoothness on microscopic scales, shocks manifest as macroscopic discontinuities. This talk begins with the observation that shock formation arises from the flow map reaching the boundary of the manifold of diffeomorphisms. We modify its geometry such that geodesics approach but never reach the boundary. The resulting information geometric regularization (IGR) has smooth solutions while avoiding the excessive dissipation of viscous regularizations, accelerating and simplifying the simulation of flows with shocks. We prove the existence of global strong IGR solutions in the unidimensional pressureless case and illustrate its practical utility on multidimensional examples with complex shock interactions. With S. Bryngelson and other collaborators, we use IGR to conduct the first compressible flow simulation exceeding a quadrillion degrees of freedom. The modified geometry of the diffeomorphism manifold is the information geometry of the mass density. The last part of the talk explains how this observation motivates information geometric mechanics that views the solutions of continuum mechanical PDEs as parameters of probability distributions originating from statistical physics. Replacing the Euclidean geometry of individual particles with the information geometry of statistical families promises performant numerical methods that preserve the positivity of densities and energies and readily integrate with scientific machine learning. About the speaker: Florian Schäfer is an assistant professor at the Courant Institute of Mathematical Sciences at New York University. Prior to joining Courant, he was an assistant professor at Georgia Tech and received his PhD in applied and computational mathematics at Caltech, working with Houman Owhadi. Before that, he received bachelor's and master's degrees in mathematics at the University of Bonn. His research addresses fundamental problems in computational science and engineering using insights from statistics and geometry. It was awarded a SIGGRAPH best technical paper award, a Gordon Bell Prize finalist designation, and a Sloan Research Fellowship in Mathematics. Up next: Thursday, April 9 – Daniel B. Szyld (Temple). There are a few slots to meet with the speakers, please let us know if you would like one (gilles@princeton.edu). - Liza Rebrova, Marc Aurèle Gilles, and Jorge Garza Vargas.