Counting the number of stationary solutions of partial differential equations via infinite dimensional sampling.

This paper is concerned with the problem of counting solutions of stationary nonlinear Partial Differential Equations (PDEs) when the PDE is known to admit more than one solution. We suggest tackling the problem via a sampling-based approach. The method allows one to find solutions that are stable, in the sense that they are stable equilibria of the associated time-dependent PDE. We test our proposed methodology on the McKean-Vlasov PDE, more precisely on the problem of determining the number of stationary solutions of the McKean-Vlasov equation.This article is part of the theme issue 'Partial differential equations in data science'.