Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space

Abstract: We compute the ground state $u$ of the Gross--Pitaevskii equation (GPE) via Wasserstein gradient descent in diffeomorphism space. We represent the density $ρ=u^2$ as the push-forward of a fixed reference measure through a parameterized transport map $T_θ$, realized by a boundary-preserving Neural ODE. The Wasserstein gradient flow on probability densities then lifts to natural gradient descent in the finite-dimensional parameter space, with metric tensor given by the pullback of the Wasserstein metric. The method is entirely mesh-free and preserves the unit-mass constraint without normalization. We present numerical experiments in dimensions $d=1,2,3$ and demonstrate that the parameterized Wasserstein gradient flow (PWGF) output can be used to initialize the $H^1$ Sobolev gradient flow, reducing the initial energy gap by a factor of $7$ in 2D and $4.5$ in 3D compared to trivial initial conditions.