We construct a neural network to learn the Reissner-Nordström-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.

中文翻译：

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从全息电导率学习黑洞度量

我们构建了一个神经网络，以基于全息术的光导率数据来学习 Reissner-Nordström-anti–de Sitter 黑洞度量。根据光导率重写麦克斯韦场的线性微扰方程，从而基于该微分方程的离散化构建神经网络。与之前所有反德西特/深度学习对偶模型相比，度量函数的导数出现在运动方程中，我们提出了不同的有限差分方法来离散化该函数。还提出了降低电导率的概念，以避免地平线附近的光导率发散。详细研究了训练结果对截止位置、温度以及频率范围的依赖性。