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arXiv · 2603.13579

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

Xiangxiong Zhang·Haomin Zhou
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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.

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Xiangxiong Zhang, Haomin Zhou. 2026-03-13. Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space. https://arxiv.org/abs/2603.13579

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