AdS/Deep-Learning made easy II: neural network-based approaches to holography and inverse problems

TL;DR

The paper demonstrates physics-informed machine learning as a robust framework for solving inverse problems in holography and holographic condensed matter. By leveraging Neural ODEs and PINNs, it shows how boundary data such as QCD EoS and T-linear resistivity can be used to recover bulk potentials and couplings in AdS/CFT models, and it strengthens the connection between holography and classical mechanics through intuitive analogies. It also introduces Kolmogorov-Arnold Networks as a more stable, efficient alternative in certain settings and outlines a systematic pathway to scale these methods, extend them with Bayesian or XPINN variants, and apply them to real-world holographic contexts. Collectively, the work provides a comprehensive, cross-disciplinary framework for data-driven discovery and model reconstruction in high-energy physics, condensed matter, and beyond.

Abstract

We apply physics-informed machine learning (PIML) to solve inverse problems in holography and classical mechanics, focusing on neural ordinary differential equations (Neural ODEs) and physics-informed neural networks (PINNs) for solving non-linear differential equations of motion. First, we introduce holographic inverse problems and demonstrate how PIML can reconstruct bulk spacetime and effective potentials from boundary quantum data. To illustrate this, two case studies are explored: the QCD equation of state in holographic QCD and $T$-linear resistivity in holographic strange metals. Additionally, we explicitly show how such holographic problems can be analogized to inverse problems in classical mechanics, modeling frictional forces with neural networks. We also explore Kolmogorov-Arnold Networks (KANs) as an alternative to traditional neural networks, offering more efficient solutions in certain cases. This manuscript aim to provide a systematic framework for using neural networks in inverse problems, serving as a comprehensive reference for researchers in machine learning for high-energy physics, with methodologies that also have broader applications in mathematics, engineering, and the natural sciences.

Figures (17)

• Figure 1: A schematic architecture of a deep neural network with two hidden layers. Each weight $W^{(ij)}_m$ connects neuron $x^{(i)}_{m-1}$ in layer $\mathcal{N}_{m-1}$ to neuron $x^{(i)}_m$ in $\mathcal{N}_m$. Here, $x^{(a)}_b$ means the $a$-th neuron in the $b$-th layer.
• Figure 2: Schematic representation of the model architecture (left) and training workflow of the neural ODE framework (right). The deep neural network, parameterized by $\theta$, predicts the model function $V(\phi)$, which is used in the ODE solver to compute $\phi(z)$. The output is compared with data to form the loss, and the parameters $\theta$ are updated via optimization.
• Figure 3: Schematic illustration of the PINN framework. The solution $\phi(z)$ and the potential $V(\phi)$ are represented by the deep neural networks parameterized by $\theta$ and $\bar{\theta}$. The network outputs and their derivatives enter the physical loss, which enforces the equations of motion.
• Figure 4: Speed of sound $C_S^2$ (left) and equation of state $s(T)$ (right) for the potential $V(\phi)$ in \ref{['FirstPotential']}. The critical temperature is around $T_c\approx0.18$, and at high temperatures, $C_S^2$ approaches $1/3\approx 0.33$, as expected for conformal behavior.
• Figure 5: The trained potential $V(\phi)$ from the QCD equation of state, Fig. \ref{['FIG:EoS_data']}, using PINNs. The true solution is \ref{['FirstPotential']}.