Deep learning and the AdS/CFT correspondence

Abstract

We present a deep neural network representation of the AdS/CFT correspondence, and demonstrate the emergence of the bulk metric function via the learning process for given data sets of response in boundary quantum field theories. The emergent radial direction of the bulk is identified with the depth of the layers, and the network itself is interpreted as a bulk geometry. Our network provides a data-driven holographic modeling of strongly coupled systems. With a scalar φ 4 theory with unknown mass and coupling, in unknown curved spacetime with a black hole horizon, we demonstrate our deep learning (DL) framework can determine them which fit given response data. First, we show that, from boundary data generated by the AdS Schwarzschild spacetime, our network can reproduce the metric. Second, we demonstrate that our network with experimental data as an input can determine the bulk metric, the mass and the quadratic coupling of the holographic model. As an example we use the experimental data of magnetic response of a strongly correlated material Sm0.6Sr0.4MnO3. This AdS/DL correspondence not only enables gravity modeling of strongly correlated systems, but also sheds light on a hidden mechanism of the emerging space in both AdS and DL. FIG. 1: The AdS/CFT and the DL. Top: a typical view of the AdS/CFT correspondence. The CFT at a finite temperature lives at a boundary of asymptotically AdS spacetime with a black hole horizon at the other end. Bottom: a typical neural network of a deep learning.horizon condition. Therefore, a successful machine learning results in a concrete metric of a holographic modeling of the system measured by the experiment [67]. We call this implementation of the holographic model into the deep neural network as AdS/DL correspondence.We check that the holographic DL modeling nicely arXiv:1802.08313v1 [hep-th]

Deep learning the holographic black hole with charge

“…Taking 1000 positive and 1000 negative data points forms the data set. This 2000 data satisfy the distribution as shown in Fig.3, which is comparable with the results of the Schwarzschild case in [16].…”

“…As a contrast, the label is called the target value. The difference between actual value and target value is the loss, which is described by the loss function [16]…”

“…Then the neural network transports the data from input layer to the final (η f in ). Following[16], choosing f (F ) = [tanh(100(F − 0.1) − tanh(100(F + 0.1)) + 2]/2(19) as the activation function is effective. Here F is one of final layer neuron(in the limit η f in → 0, equation(18)is equivalent to F≡ π(η f in )).…”

AdS/CFTcorrespondence as a deep Boltzmann machine

“…This means that the saddle point of the deep Boltzmann machine brings it to a folded feed-forward type deep neural network. The AdS/CFT interpretation of a feed-forward neural network was studied in [15,16] and the trained weights exhibit an interesting physical picture.…”

“…In [14], entanglement feature of a free fermion chain was trained at a random tensor network as a deep Boltzmann machine. The holographic interpretation for feed-foward deep neural networks was proposed and studied in [15,16] for training QFT and QCD linear response functions, and the obtained emergent bulk spacetime for large N c QCD exhibits interesting physical properties and computes other observables as predictions. While our work here naturally relates to these work, in this paper we concentrate on deep Boltzmann machines as an AdS/CFT correspondence.…”