Efficient numerical methods for Bose-Einstein condensates

In this project we study numerical methods for the Gross-Pitaevskii eigenvalue problem with angular momentum, which is used to model rotating Bose-Einstein condensates (BECs). One of the most prominent examples of BECs is superfluids; fluids with zero viscosity. A key property of superfluids is their ability to form quantized vortices when rotated.

The diameter of such a vortex is small compared to the overall domain, meaning that the vortex pattern forms a fine detailed structure of multiscale type.

To model this efficiently, we suggest a generalized finite element method based on techniques from Localized Orthogonal Decomposition (LOD) where the solution space is split into a coarse (low dimensional) space and a fine detail space. The main challenge is to construct the split so that the vortices are resolved efficiently in the coarse space.

We shall focus on theoretical aspects such as well-posedness and error estimates, as well as practical issues such as implementation. Furthermore, we are interested in performing bifurcation studies as one or more parameters are varied. This is a numerically challenging task, since a change in the parameter often means recomputing the LOD space.

To resolve this, we turn to machine learning to approximate the solution operator.

We aim to investigate techniques based on the general idea of physics informed neural networks, where the network is penalized to follow the underlying partial differential equation.