Eur. Phys. J. C (2025) 85: 63
https://doi.org/10.1140/epjc/s10052-025-13759-0

Regular Article - Theoretical Physics

Neural ODEs for holographic transport models without translation symmetry

¹ Department of Physics, Shanghai University, 200444, Shanghai, China
² School of Physics, University of Chinese Academy of Sciences, 100049, Beijing, China
³ Center for Gravitation and Cosmology, Yangzhou University, 225009, Yangzhou, China

^a sfwu@shu.edu.cn

Received: 16 August 2024
Accepted: 29 December 2024
Published online: 24 January 2025

Abstract

We investigate the data-driven holographic transport models without translation symmetry, focusing on the real part of frequency-dependent shear viscosity, ${\eta }_{\text{re}}(\omega )$. We develop a radial flow equation of the shear response and establish its relation to ${\eta }_{\text{re}}(\omega )$ for a wide class of holographic models. This allows us to determine ${\eta }_{\text{re}}(\omega )$ 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. By specifying the metric to be asymptotically AdS, we also train the Neural ODE to learn the metric and mass simultaneously. Furthermore, we apply the Neural ODE to the experimental data from a two-dimensional liquid strongly coupled dusty plasma. Interestingly, the emergent metric exhibits non-monotonic behavior, and the mass squared is negative.