Gaussian-process-regression-based method for the localization of
  exceptional points in complex resonance spectra
Patrick Egenlauf Patric Rommel Jörg Main
Resonances in open quantum systems depending on at least two controllable
parameters can show the phenomenon of exceptional points (EPs), where not only
the eigenvalues but also the eigenvectors of two or more resonances coalesce.
Their exact localization in the parameter space is challenging, in particular
in systems, where the computation of the quantum spectra and resonances is
numerically very expensive. We introduce an efficient machine learning
algorithm to find exceptional points based on Gaussian process regression
(GPR). The GPR-model is trained with an initial set of eigenvalue pairs
belonging to an EP and used for a first estimation of the EP position via a
numerically cheap root search. The estimate is then improved iteratively by
adding selected exact eigenvalue pairs as training points to the GPR-model. The
GPR-based method is developed and tested on a simple low-dimensional matrix
model and then applied to a challenging real physical system, viz., the
localization of EPs in the resonance spectra of excitons in cuprous oxide in
external electric and magnetic fields. The precise computation of EPs, by
taking into account the complete valence band structure and central-cell
corrections of the crystal, can be the basis for the experimental observation
of EPs in this system.