Deep learning method for testing the cosmic distance duality relation

The cosmic distance duality relation (DDR) is constrained by a combination of type-Ia supernovae (SNe Ia) and strong gravitational lensing (SGL) systems using the deep learning method. To make use of the full SGL data, we reconstruct the luminosity distance from SNe Ia up to the highest redshift of SGL using deep learning, and then, this luminosity distance is compared with the angular diameter distance obtained from SGL. Considering the influence of the lens mass profile, we constrain the possible violation of the DDR in three lens mass models. The results show that, in the singular isothermal sphere and extended power-law models, the DDR is violated at a high confidence level, with the violation parameter begin{document}$ eta_0=-0.193^{+0.021}_{-0.019} $end{document} and begin{document}$ eta_0=-0.247^{+0.014}_{-0.013} $end{document}, respectively. In the power-law model, however, the DDR is verified within a 1σ confidence level, with the violation parameter begin{document}$ eta_0=-0.014^{+0.053}_{-0.045} $end{document}. Our results demonstrate that the constraints on the DDR strongly depend on the lens mass models. Given a specific lens mass model, the DDR can be constrained at a precision of begin{document}$O(10^{-2}) $end{document} using deep learning.