Learning the black hole metric from holographic conductivity

Abstract

We construct a neural network to learn the Reissner-Nordstr\"om-anti--de Sitter black hole metric based on the data of optical conductivity by holography. The linear perturbative equation for the Maxwell field is rewritten in terms of the optical conductivity such that the neural network is constructed based on the discretization of this differential equation. In contrast to all previous models in anti--de Sitter/deep learning duality, the derivative of the metric function appears in the equation of motion and we propose distinct finite difference methods to discretize this function. The notion of the reduced conductivity is also proposed to avoid the divergence of the optical conductivity near the horizon. The dependence of the training outcomes on the location of the cutoff, the temperature as well as the frequency range is investigated in detail. This work provides a concrete example for the reconstruction of the bulk geometry with the given data on the boundary by deep learning.