Counting the number of stationary solutions of Partial Differential Equations via infinite dimensional sampling

TL;DR

This paper introduces a sampling-based method to count stationary solutions of nonlinear PDEs, demonstrated on the McKean-Vlasov equation, addressing the challenge of multiple solutions.

Contribution

It proposes a novel sampling approach for counting solutions of stationary nonlinear PDEs, with application to the McKean-Vlasov equation.

Findings

Successfully counts stationary solutions of the McKean-Vlasov PDE

Demonstrates effectiveness of sampling approach for nonlinear PDEs

Provides a new tool for solution enumeration in complex PDEs

Abstract

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. 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 (or porous medium) equation.