Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space
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| Format: | Preprint |
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2026
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| _version_ | 1866915862013804544 |
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| author | Zhang, Xiangxiong Zhou, Haomin |
| author_facet | Zhang, Xiangxiong Zhou, Haomin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13579 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space Zhang, Xiangxiong Zhou, Haomin Numerical Analysis 35Q55, 49Q22, 65K10, 81Q05 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. |
| title | Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space |
| topic | Numerical Analysis 35Q55, 49Q22, 65K10, 81Q05 |
| url | https://arxiv.org/abs/2603.13579 |