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

TL;DR

This paper introduces a mesh-free method using Wasserstein gradient flow in diffeomorphism space, employing neural ODEs to compute the ground state of the Gross-Pitaevskii equation efficiently.

Contribution

It develops a novel neural ODE-based approach for Wasserstein gradient descent in diffeomorphism space to find GPE ground states without mesh reliance.

Findings

Method reduces initial energy gap significantly in 2D and 3D.

Approach is mesh-free and preserves mass constraint inherently.

Numerical experiments confirm effectiveness across dimensions.

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 $d=1,2,3$ 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.