Gravitational Duals from Equations of State

TL;DR

The paper addresses the inverse holography problem of reconstructing a five-dimensional gravitational theory from a prescribed gauge theory equation of state S(T). It introduces a physics-informed neural network framework with a dual-network, solution-bundle architecture to learn the scalar potential V(φ) and the corresponding black-brane geometry from S(T). Across crossover, and first-/second-order phase transitions, the method achieves sub-percent accuracy in V(φ) and a few-percent accuracy in S(T), while revealing the role of multi-scale and multi-entangled geometry in the learning process. This approach generalizes to more complex gravity theories and offers a data-driven path to studying non-equilibrium dynamics via holography.

Abstract

Holography relates gravitational theories in five dimensions to four-dimensional quantum field theories in flat space. Under this map, the equation of state of the field theory is encoded in the black hole solutions of the gravitational theory. Solving the five-dimensional Einstein's equations to determine the equation of state is an algorithmic, direct problem. Determining the gravitational theory that gives rise to a prescribed equation of state is a much more challenging, inverse problem. We present a novel approach to solve this problem based on physics-informed neural networks. The resulting algorithm is not only data-driven but also informed by the physics of the Einstein's equations. We successfully apply it to theories with crossovers, first- and second-order phase transitions.

Figures (15)

• Figure 1: Equation of state for a gauge theory with a crossover ($\phi_M=5$), a second-order phase transition ($\phi_M=1.08$), and a first-order phase transition ($\phi_M=1$) Bea:2018whf. In the latter two cases, the critical temperatures are indicated by solid vertical lines. The entropy density is shown in units of the intrinsic scale in the theory (top) and in units of the temperature (bottom).
• Figure 2: Potentials that give rise to the equations of state in Fig. \ref{['SofT']}Bea:2018whf. All potentials share the same maximum at $\phi=0$, but they have minima at different positions indicated by the solid vertical lines.
• Figure 3: Structure of the implemented NN-system of two NNs with Gaussian localization. The red and blue arrows represent the backpropagation process, in which the parameters $W_j$ ($j=D,V$) are modified such that the NN's state in parameter space flows in the direction of the gradient of the loss function, $\nabla_{W_j}L$.
• Figure 4: Sampling of the $S(T)$ curve for a crossover (top), a second-order transition (middle) and a first-order transition (bottom), in the range of temperatures $0$ to $0.7$. The points along the curves are sampled more densely around the phase transition or crossover ($T/\Lambda \approx 0.4$), as well as in the IR ($T/\Lambda \approx 0$).
• Figure 5: Loss function (squared residual from our ODEs) in log scale as a function of the epoch, for the best run of the first order transition case ($\phi_M=1$). This NN has been trained for 3.5 million epochs. The minimum loss achieved is $\approx7.4\cdot 10^{-7}$.