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Deep Learning and AdS/CFT
Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, Akio Tomiya
TL;DR
The paper asks whether boundary response data can be modeled holographically when the bulk metric is unknown. It uses a deep neural network to infer that metric and demonstrates the approach on AdS Schwarzschild data and experimental magnetic-response data.
Problem
Holographic modeling has succeeded mainly for systems with manifest symmetries, while whether a given quantum system has a gravity dual and how to construct its model remain unknown.
Source: p. 1Method
A deep neural network represents the curved-spacetime scalar equation, with layer depth encoding the radial direction and trained weights encoding a metric component, to infer a bulk metric from boundary response data assuming a black hole horizon.
Source: p. 1Results
m2L2 = 5.6 ± 2.5 and λ/L = 0.61 ± 0.22; experimental magnetic-response data yielded a learned metric approaching a constant at the AdS boundary.
Source: pp. 2, 4Takeaways & Limitations
The authors state that vector fields could let the framework determine additional bulk metric components beyond h(η).
Source: p. 4The 155K analysis may not be near criticality, a classical bulk scalar description is not established for the material, and no suitable energy condition constrains h(η).
Source: BLACK HOLE METRIC AND COORDINATE SYSTEMS · p. 11, COMMENTS ON THE CONFORMAL DIMENSIONS · p. 14
Abstract
from arXiv · showhide
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 $\phi^4$ 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 Sm$_{0.6}$Sr$_{0.4}$MnO$_3$. 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.
HAMILTONIAN SYSTEMS REALIZED BY DEEP NEURAL NETWORK
A direct identification of network units with canonical variables cannot represent nonlinear Hamilton equations, but an expanded identification yields a network representation that includes the paper’s curved-space scalar field equation.
HAMILTONIAN SYSTEMS REALIZED BY DEEP NEURAL NETWORK
- The direct mapping of units to q and p and layer propagation to time evolution permits only free, linear Hamilton equations.The authors conclude that this simple identification does not allow nonlinear activation functions.Source: HAMILTONIAN SYSTEMS REALIZED BY DEEP… · p. 8
- The improved construction adds units x0 and x3 and treats successive linear, local nonlinear, and linear transformations as one time translation.The adjacent layer sets combine into a single linear transformation, retaining the standard deep-network form.Source: HAMILTONIAN SYSTEMS REALIZED BY DEEP… · p. 8
- Under the symplectic constraint, the improved construction yields a generic class of nonlinear Hamiltonians representable by a deep neural network.The scalar field equation in curved geometry is within this class.Source: HAMILTONIAN SYSTEMS REALIZED BY DEEP… · p. 9
- The authors leave open whether a more involved identification of time translation and layer propagation can accommodate Hamiltonians outside the form in (S.21).They defer the generic argument to future investigation.Source: HAMILTONIAN SYSTEMS REALIZED BY DEEP… · p. 9
ERROR FUNCTION OF THE ADS SCALAR SYSTEM
The section relates learning updates to a gradient flow and examines where metric updates act along the emergent radial direction, distinguishing the explicit λ = 0 analysis from simulations with λ ≠ 0.
ERROR FUNCTION OF THE ADS SCALAR SYSTEM
- For λ = 0, the loss can be explicitly expressed using a path-ordered solution of the scalar-field equation.The learning process is equivalent to a gradient-flow equation with fictitious time τ.Source: ERROR FUNCTION OF THE ADS SCALAR SYSTEM · pp. 9–10
- With the constant initial h(η), the λ = 0 update acts near both the black-hole horizon and the AdS boundary.The update contains terms proportional to exp[(λ+ − λ−)η] and exp[−(λ+ − λ−)η].Source: ERROR FUNCTION OF THE ADS SCALAR SYSTEM · p. 10
- For simulations with λ ≠ 0, backpropagation is expected to be suppressed near the input layer, unlike the explicit λ = 0 case.The authors attribute the difference to the trivial activation function assumed to obtain the explicit loss at λ = 0.Source: ERROR FUNCTION OF THE ADS SCALAR SYSTEM · p. 10
BLACK HOLE METRIC AND COORDINATE SYSTEMS
The section sets the radial coordinate convention and describes the generic behavior of the metric function, while noting that no suitable energy condition was found to constrain its evolution.
BLACK HOLE METRIC AND COORDINATE SYSTEMS
- The radial coordinate places the AdS boundary at η = ∞ and the black-hole horizon at η = 0.The coordinate transformation brings the AdS Schwarzschild metric into the form used for the scalar-field equation.Source: BLACK HOLE METRIC AND COORDINATE SYSTEMS · p. 10
- Generically, h(η) behaves as approximately 1/η near the horizon and approaches a constant proportional to the AdS radius L at the boundary.The passage states this behavior regardless of temperature or matter energy-momentum tensor.Source: BLACK HOLE METRIC AND COORDINATE SYSTEMS · p. 11
- The authors found no suitable energy condition to constrain h(η), so non-monotonic metric functions are allowed among learned metrics.Source: BLACK HOLE METRIC AND COORDINATE SYSTEMS · p. 11
Comments on the regularization
Regularization selects smooth metric profiles among fits with comparable accuracy, supporting a continuum-limit interpretation. In the experimental-data runs, changing its strength changes the learned near-horizon metric while reported loss remains about 0.02.
Comments on the regularization
- The regularization selects smooth metrics among learned profiles with similar accuracy, rather than spoiling the learning process.Smoothness is preferred because the authors seek a continuum limit for h(η).Source: Comments on the regularization · p. 11
- For the Schwarzschild reconstruction, 50 trials learn the asymptotic AdS region well, while near-horizon profiles are qualitatively right but quantitatively deviate, possibly because of discretization.The caption notes that an unregularized profile also achieves high accuracy, although the text says such profiles can be far from the Schwarzschild metric.Source: Comments on the regularization · p. 11
- For experimental-data fits, the regularization includes a term that fits h(η) near the horizon to the expected 1/η behavior.The stated regularization value for this term is 10^-4.Source: Comments on the regularization · p. 11
- Across the experimental-data cases, learned metrics achieve a loss of approximately 0.02 despite differing in how regularization selects physically sensible profiles.The authors use an intermediate regularization setting for Fig. 6 because the near-horizon deviation is comparable to that near the asymptotic region.Source: Comments on the regularization · p. 11
Numerical experiment 1: Reconstructing AdS Schwarzschild black hole
The first numerical experiment tests whether the network can reconstruct an AdS Schwarzschild metric from boundary-field data. Its setup generates labeled examples by propagating sampled boundary conditions and trains the network on those examples.
Numerical experiment 1: Reconstructing AdS Schwarzschild black hole
- The experiment aims to test reconstruction of the AdS Schwarzschild black-hole metric, using a network whose learned parameters are h(η(n)).The setup specifies a scalar field with fixed mass and quartic coupling, but the supplied passage does not include their values.Source: Numerical experiment 1: Reconstructing… · pp. 11, 13
- Training examples pair boundary values (φ, π) with a binary signal indicating whether propagation toward the horizon satisfies the consistency criterion.Samples are propagated using the equation of motion in the AdS Schwarzschild metric; the passages define the signal using a threshold of 0.1.Source: Numerical experiment 1: Reconstructing… · p. 12
- The data collection procedure balances positive and negative examples by sampling further from the category that has fewer examples.The resulting categories are concatenated for training.Source: Numerical experiment 1: Reconstructing… · pp. 12–13
- Before training, h(η(n)) is initialized from a Gaussian N(1/η(n), 1); training data are split into 200 mini-data sets, updated with Adam, and iterations stop when target loss falls below 0.0002.The figure caption describes a learned metric as high-accuracy without regularization Ereg and says the setup matches the AdS Schwarzschild metric reproduction.Source: Figure 8 · p. 12, Numerical experiment 1: Reconstructing… · p. 13
Numerical experiment 2: Emergent metric from experimental data
The second experiment uses magnetization measurements for Sm0.6Sr0.4MnO3 at 155 K to construct training data for inferring a bulk metric and scalar-field parameters. A polynomial fit to the measurements supplies the reference response curve.
Numerical experiment 2: Emergent metric from experimental data
- The experimental input consists of magnetic field H and magnetization M measurements for Sm0.6Sr0.4MnO3 at 155 K, with a 15th-order polynomial fit f(H) used to pad the data.The measurements are plotted as paired values before fitting.Source: Numerical experiment 2: Emergent… · p. 13
- Training inputs are sampled H and M values, and the answer signal classifies whether M lies within a noise band around f(H).The procedure prepares 10^4 positive and 10^4 negative examples.Source: Numerical experiment 2: Emergent… · pp. 13–14
- The 10-layer network trains h(η(n)), α, β, m2, and λ together, stopping when the target loss falls below 0.02.Initial values are sampled from the distributions specified for each parameter in the passage.Source: Numerical experiment 2: Emergent… · p. 14
COMMENTS ON THE CONFORMAL DIMENSIONS
The section relates the magnetic critical exponents to a bound on the scaling dimension, then compares that bound with the value inferred from the material data. The inferred dimension exceeds the bound, and the authors identify modeling and setup factors that warrant caution.
COMMENTS ON THE CONFORMAL DIMENSIONS
- For the 155 K data, the inferred conformal dimension is ∆+ = 4.89 ± 0.32, exceeding the stated D = 3 bound ∆+ < 3.The authors caution against straightforward interpretation of this value.Source: COMMENTS ON THE CONFORMAL DIMENSIONS · p. 14
- The authors note that the scalar model does not properly represent the operator's spin structure, and that conformal-dimension identification can differ across holographic models.They cite several proposed holographic treatments of magnetization.Source: COMMENTS ON THE CONFORMAL DIMENSIONS · p. 14
- The numerical dimension uses ηini = 1 to reduce computation; choosing ηini/L much larger would have made the bound violation milder.Source: COMMENTS ON THE CONFORMAL DIMENSIONS · p. 14
- The authors also question whether 155 K is close enough to criticality and whether the scaling law applies, since the transition order is unclear.They further state there is no persuasive reason to assume this material is described holographically by a classical bulk scalar field.Source: COMMENTS ON THE CONFORMAL DIMENSIONS · p. 14