Neural ODEs for holographic transport models without translation symmetry

We investigate the data-driven holographic transport models without translation symmetry, focusing on the real part of frequency-dependent shear viscosity, $η_{\mathrm{re}}(ω)$. We develop a radial flow equation of the shear response and establish its relation to $η_{\mathrm{re}}(ω)$ for a wide class of holographic models. This allows us to determine $η_{\mathrm{re}}(ω)$ of a strongly coupled field theory by the black hole metric and the graviton mass. The latter serves as the bulk dual to the translation symmetry breaking on the boundary. We convert the flow equation to a Neural Ordinary Differential Equation (Neural ODE), which is a neural network with continuous depth and produces output through a black-box ODE solver. Testing the Neural ODE on three well-known holographic models without translation symmetry, we demonstrate its ability to accurately learn either the metric or mass when given the other. Additionally, we illustrate that the learned metric can be used to predict the derivative of entanglement entropy $S$ with respect to the size of entangling region $l$.