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.

Figures (7)

• Figure 1: 1D PWGF ground-state computation. Top-left: energy error $|E^k-E^*|$ (log scale). Top-right:$L^2$ error $\|u^k-u^*\|$; decreases to the floor $\approx 0.036$ (Table \ref{['tab:errorsummary']}). Bottom-left: gradient norm $\|\nabla_\theta E^k\|$. Bottom-right: reconstructed $u_\theta$ vs. exact $u^*$.
• Figure 2: Plateau $\|u^k-u^*\|_{L^2}$ vs. each hyperparameter (baseline: $N=3000$, $H=10$, $N_{\rm ODE}=10$; dashed line at $0.036$). (a)$N$: non-monotone; $N=3000$ is optimal. (b)$H$: saturates at $H=10$. (c)$N_{\rm ODE}$: negligible improvement ($0.0364\to0.0354$ for $8\times$ more compute).
• Figure 3: 2D problem ($\beta=10$, $n=200$). Top: potential $V$, PWGF warm start $u_\theta$ ($E_{\rm FD}=0.2327$), and H1 reference $u^*_h$ ($E^*_h=0.21706$). Bottom left: PWGF energy history. Bottom center/right: energy and eigenvalue error vs. H1 iteration.
• Figure 4: 3D GPE ($\beta=1600$, $n=199$). (a)--(b) Isosurfaces at level $0.002$: PWGF solution ($E=55.26$) and reference solution ($E^*_h=33.794$, $3\times3\times3$ blobs). (c)--(d) Slices at $z\approx-4,0,+4$; peak $\approx0.221$.
• Figure 5: 3D Beta(5,5)$^3$ density $\mu(\mathbf{z})=C_1^3\prod_{k=1}^3(1-z_k^2/L^2)^4$ on $(-8,8)^3$, displayed with the same isovalue and colorbar as Figure \ref{['fig:3d-visualization-n199']}. Left: isosurface at level $0.002$ — a single smooth blob centered at the origin, in contrast to the $3\times3\times3$ multi-blob structure of the true ground state (Figure \ref{['fig:3d-visualization-n199']}). Right: slices at $z\approx-4,0,+4$.

Theorems & Definitions (4)

• Remark 1: Regularization in grid-based density methods
• Remark 2: No regularization needed in PWGF
• Remark 3
• Remark 4: Energy below $E^*$