Computing the Gross-Pitaevskii Ground State via Wasserstein Gradient Flow in Diffeomorphism Space
This provides a more efficient initialization method for solving a specific quantum physics equation, but it is incremental as it builds on existing gradient flow techniques.
The paper tackles computing the ground state of the Gross-Pitaevskii equation by using a Wasserstein gradient flow in diffeomorphism space, resulting in a mesh-free method that reduces the initial energy gap by factors of 7 in 2D and 4.5 in 3D compared to trivial initializations.
We compute the ground state $u$ of the Gross--Pitaevskii equation (GPE) via Wasserstein gradient descent in diffeomorphism space. We represent the density $Ï=u^2$ 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.