Solvers, Models, Learners: Statistical Inspiration for Scientific Computing

The convergence of scientific computing with statistics and machine learning is an exciting development in scientific computing. In this talk, I will present two lines of work that blur the line between statistical inference and numerical computation. The first part of the talk presents state-of-the-art algorithms for solving elliptic PDEs by interpreting them as Gaussian processes and exploiting their conditional independence properties. This approach allows the efficient learning of elliptic solution operators from solution pairs, with promising empirical results on fractional order PDEs and empirical closure models of turbulent flows.

The second part of the talk discusses how to mitigate the formation of shock singularities in the barotropic Euler equations using an inviscid regularization. This work combines the seminal work of Vladimir Arnold on geometric hydrodynamics with ideas from interior point methods for positive definite programming and the information geometry of Amari and Chentsov.