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 $\rho =u^2$ as the push-forward of a fixed reference measure through a parameterized transport map $T_{\theta }$, 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 $d=1$ to $4$, with separable and non-separable potentials, using a product transport map in the separable case and a full map with coupled components in the non-separable one. The tests suggest that the output of the parameterized Wasserstein gradient flow (PWGF) can be used as an effective warm start for conventional solvers such as the $H^1$ Sobolev gradient flow.