Learning the Position of Image Vortices from Data

TL;DR

This work addresses learning the effective boundary-induced image-vortex dynamics for a single vortex in Bose-Einstein condensates confined by circular power-law traps. It harnesses Sparse Identification of Nonlinear Dynamics (SINDy), including its implicit/rational-function variant, to extract a sparse, data-driven implicit equation that incorporates a single image vortex whose distance parameter $\varphi$ adapts across trap shapes. Through synthetic hard-wall data, Gross-Pitaevskii equation simulations, and ensemble-SINDy analyses, the study shows that a single, well-placed image vortex suffices to describe the vortex precession in harmonic and higher-power traps, with $\varphi^2$ approaching the hard-wall limit as the boundary becomes sharper. The resulting framework aligns well with experimental observations and offers a scalable, data-driven path to learn vortex-boundary interactions in complex geometries, with future extensions to dissipation and novel trap architectures.

Abstract

The point vortex model is an idealized model for describing the dynamics of many vortices with numerical efficiency, and has been shown to be powerful in modeling the dynamics of vortices in a superfluid. The model can be extended to describe vortices in fluids with a well defined boundary, as an image vortex can be added to the equations of motion to impose the correct velocity profile at the boundary. The mathematical formulation of the image vortex depends on the boundary in question, and is well known for a wide variety of problems, although the formulation of an image vortex in a fluid with a soft boundary remains under debate, as the boundary condition is ill-posed. Such a boundary is common-place in the dynamics of a vortex in an ultra-cold atomic Bose-Einstein condensate, for example, which is typically trapped in a harmonic potential. In order to address this problem, the Sparse Identification of Nonlinear Dynamics framework is applied to data from mean-field simulations to extract an approximate point vortex model for a vortex in a circular power law trapping potential. A formulation for the position of an image vortex in such a trap is presented, and the accuracy of this model is evaluated.

Figures (6)

• Figure 1: Applying the SINDy algorithm to synthetic data for a hard walled trap. Top panel: the absolute value of the returned coefficients for the $x$ (blue squares) and $y$ (red circles) equations of motion. The correct coefficients from Eqn. \ref{['eqn:Hard_Wall']} are indicated with asterisks. Bottom left: increasing the thresholding parameter $\lambda$ reduces the number of non-zero terms in the coefficient vector. Bottom right: the model error is calculated, the lowest error combined with the most parsimonious model gives the best choice of model coefficients.
• Figure 2: The absolute value of the returned value of the coefficients for the $x$ (blue squares) and $y$ equations of motion of a single vortex in a hard wall trap. These coefficients are for the extended library of candidate functions given in Eqn. \ref{['eqn:synthetic_extended_library']}, which now include cubic terms. The correct terms in the model, indicated by asterisks, are identified by the SINDy algorithm even with this extended library
• Figure 3: Applying the SINDy algorithm to GPE data of a single vortex in a harmonic trap. Top panel: the absolute value of the returned coefficients for the $x$ (blue squares) and $y$ (red circles) equations of motion; the dashed line $R^{-2}$ is added as a guide to the eye. Bottom left: increasing the thresholding parameter reduces the number of non-zero terms. Bottom right: the error is calculated for models that contain the given number of terms.
• Figure 4: Application of Ensemble SINDy to the data obtained from the GPE with harmonic trapping. First (third) row: the inclusion probability of terms in the $\dot{X}$ ($\dot{Y}$) equations of motion. Second (fourth) row: normalised histogram and probability density function of the fitted normal distribution for the value of the coefficients of the terms $y$, $\dot{X}$, $\dot{X}x^2$ and $\dot{X}y^2$ ($x$, $\dot{Y}$, $\dot{Y}x^2$ and $\dot{Y}y^2$).
• Figure 5: The position of the image vortex as a function of the trap power, $p$, can be estimated from the parameter $\varphi^2$. Blue squares indicate this for each of the values of $p$ considered, while the red shaded area indicates one standard deviation from the mean. The dashed black line indicates the hard wall limit, $\varphi^2\to R^2$.