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

TL;DR

This work tackles counting stable stationary solutions of nonlinear PDEs by introducing a sampling-based approach that leverages the invariant measure of an additive SPDE obtained by perturbing the time-dependent problem. The McKean–Vlasov PDE serves as the testbed, where the number of stable stationary states corresponds to invariant-measure modes under appropriate noise. Numerical realization uses a spectral Galerkin spatial discretization coupled with Euler–Maruyama time stepping, resolving $J$ Fourier modes and enabling efficient FFT-based computations. Simulations demonstrate metastable switching and, depending on parameters, the existence of two stable equilibria (and one unstable) or a unique bimodal stationary state, validating the method as a flexible tool for landscape exploration in infinite dimensions.

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.

Figures (7)

• Figure 1: A simple scalar multiwell potential, $U$, and the associated trajectory of \ref{['Langevin']}. As expected, the trajectory tends to persist near one of the two minima.
• Figure 2: Heat map of the solution $u_t$ to the SPDE \ref{['eqn:additiveSPDE']} with $V$ and $F$ as in \ref{['eqn:torusVF']}. From left to right, these plots have been obtained with $\sigma=0.2,0.6$ and $1$, respectively; we recall that $\sigma=1$ is above the critical threshold. The other parameter values held fixed given by: $t_{\text{max}}=3\cdot10^4,\Delta t=10^{-2}, s=0.75, \gamma = 10^{-2}$ and initial condition $u_0(x)=\frac{1}{\pi}\sin^2\,x$. On the vertical axis of each graph there is time $t$ while on the horizontal axis lies the spatial variable $x$. The colour bar is common across the three plots. These plots have been obtained by employing CairoMakie DanischKrumbiegel2021.
• Figure 3: Heat map of the solution $u_t$ to the SPDE \ref{['eqn:additiveSPDE']} with $V(x)=\cos \,4x$, and $F$ as in \ref{['eqn:torusVF']}. This plot has been obtained with the following choice of the parameter values: $\sigma=0.4, t_{\text{max}}=3\cdot10^4,\Delta t=10^{-2}, s=0.75, \gamma = 10^{-2}$ and initial condition $u_0(x)=\frac{1}{\pi}\sin^2\,x$. On the vertical axis there is time $t$ while on the horizontal axis lies the spatial variable $x$. The plots have been obtained by employing CairoMakie DanischKrumbiegel2021.
• Figure 4: Empirical verification of our algorithm's performance. We see numerical convergence in both $J$ and $\Delta t$ at the anticipated rates. Computed with $\sigma=\gamma=0.1$, $s=1$, integrated until $t_{\max}=10$. Error bars reflect 95% bootstrap percentile confidence intervals generated from $10^4$ independent trials.
• Figure 5: Eigenvalue of largest real part for the ${(m_1,m_2)}=(0,0)$ solution at different $\sigma$. In the subcritical regime, this solution is linearly unstable, while in the supercritical regime it appears stable.