Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments

Daichi Takeda; Gakuto Ogiwara; Koji Hashimoto; Koshiro Matsuo; Masaki Murata

REVIEW 4 major objections 5 minor 1 cited by

Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that a single spacetime-dependent linear response channel of a boundary theory fixes both independent components of the dual 3D black hole metric through a Runge-Kutta neural network.

desk verdict A genuinely new RK-layer architecture for holographic bulk reconstruction, but the central claim that one linear-response channel fixes the metric is under-supported by the binary-label loss and the narrow synthetic validation. read the letter →

arxiv 2411.16052 v1 pith:YFPY2D5B submitted 2024-11-25 hep-th cs.LG

Koji Hashimoto , Koshiro Matsuo , Masaki Murata , Gakuto Ogiwara , Daichi Takeda This is my paper · ORCID

classification hep-thcs.LG PACS 11.25.Tq04.60.-m

keywords AdS/CFTcorrespondencebulkreconstructioninterpretableneuralnetworkRunge-KuttalayerlinearresponseBTZblackholeemergentspacetimetabletopquantumgravity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that a single space- and time-dependent linear response measurement on a ring-shaped material contains enough information to reconstruct the three-dimensional gravitational metric dual to it under the AdS/CFT correspondence. It builds a neural network whose trainable weights are the two independent functions of the bulk metric, and whose layers simulate the Runge-Kutta integration of the bulk scalar equation of motion. Trained on synthetic boundary data labelled by whether the horizon boundary condition is satisfied, the network recovers the BTZ black hole metric and the ingoing condition at the horizon. If the method holds on real material data, it would turn linear-response measurements into a practical bulk reconstruction engine for tabletop quantum gravity experiments.

What carries the argument

The central machinery is the Runge-Kutta layer: each layer computes the fourth-order update of $(\Phi_n,\Pi_n)$ using bulk layers that evaluate $F(\xi,Z)$ from the current metric weights $\Xi(\xi)$ and $\Theta(\xi)$. Stacking 89 such layers from $\xi=0.99$ to $\xi=0.1$ turns the neural network into a discretized solver of the bulk Klein-Gordon equation. A boundary-condition layer then evaluates the combination $\xi\Pi_n+\rho_n\Phi_n$ at $\xi=0.1$, with $\rho_n=i\omega a_n+|k_n|b_n$ trained alongside the metric, so the horizon condition itself is learned. The unknown geometry is carried entirely by 178 trainable weights plus the boundary parameter, and the pure AdS3 metric provides the initial weights.

What would settle it

Train the identical architecture on linear response data generated from a different known static, rotationally symmetric metric that has the same linear response as BTZ within the limited range $|\omega|,|k_n|\le3$; if the network converges to BTZ-like profiles or fails to recover the generating metric, the claimed one-channel identifiability is false. Alternatively, take the learned $\Xi,\Theta$ and compute the linear response for held-out values such as $\omega=k_n=6$, outside the training range, and compare against the exact BTZ solution.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for a static, rotationally symmetric asymptotically AdS3 bulk, the two independent metric combinations $\Xi(\xi)=g^{\xi\xi}g_{tt}$ and $\Theta(\xi)=g^{\xi\xi}g_{\theta\theta}$ are fixed by the space- and time-dependent linear response of a single boundary scalar operator. The authors implement the bulk Klein-Gordon equation as an interpretable neural network: Runge-Kutta layers integrate the field from the AdS boundary inward, and the weights of those layers are exactly $\Xi$ and $\Theta$ after the gauge-fixing $\sqrt{-g}\,g^{\xi\xi}=C\xi^{-1}$. With 2000 synthetic examples generated from the BTZ solution and a boundary-condition loss at $\xi=0.1$, the trained weights match the BTZ profiles across $[0.1,0.99]$, and the learned boundary parameter $(a_n,b_n)=(0.50,0.01)$ reproduces the ingoing horizon condition. The authors state that this demonstrates that a single linear response channel can fix the two independent bulk metric components, generalizing earlier reconstructions that needed multiple scalar fields or nonlinear, spacetime-independent response data.

Load-bearing premise

The load-bearing premise is that enforcing the ingoing-horizon condition at a single radial location, using a finite set of synthetic training examples, is enough to single out the true metric functions across the whole radial interval; the paper gives no proof that no other metric would satisfy the same data.

Editorial extensions

If this is right

A single scalar linear-response channel suffices to constrain two independent metric components, in contrast to earlier reconstructions that required multiple scalar fields or nonlinear, spacetime-independent data.
The learned boundary parameter $(a_n,b_n)=(0.50,0.01)$ matches the ingoing BTZ horizon condition, meaning the black hole horizon emerges as a trained property of the network.
The same architecture can serve as the central engine for tabletop experiments: measuring the source and response on a ring-shaped spacetime-emergent material directly yields a candidate bulk metric.
Because the network reconstructs equation-of-motion coefficients without assuming the Einstein action, the method applies to any static, rotationally symmetric high-temperature bulk geometry for which the scalar wave equation is known.
Higher-frequency and higher-wavenumber data improve the near-horizon reconstruction of $\Theta$, whose deviation at $\xi\sim0.1$ is attributed to the limited range $|\omega|,|k_n|\lesssim3$.
The authors provide a physics-informed free-energy regularization, and report that the learned metric stays close to BTZ with or without it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the authors leave implicit is that the claimed identifiability is only demonstrated numerically on synthetic BTZ data; without a uniqueness proof, a family of metrics consistent with the same finite-data loss could remain, especially near the horizon where $\Theta$ visibly deviates.
The architecture is a general tool: replacing the BTZ data-generation step with another static, rotationally symmetric metric would test whether one linear-response channel alone can always pin down two metric functions, which is a check the paper does not perform.
The differentiable Runge-Kutta layer is a reusable physical prior for inverse problems in other settings, since it makes any unknown coefficient in an ordinary differential equation directly trainable from boundary data.
For real experimental data, the free-energy regularization discussed in Appendix C is likely to be essential, because real materials need not be governed by the Einstein equation and multiple emergent geometries could fit a limited dataset.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real step forward for ML-based bulk reconstruction, but the headline claim outruns the experiment.

First, what's new and good. The Runge-Kutta layer is a genuine improvement over the Euler discretization in earlier work, and using spacetime-dependent linear response data to recover two metric combinations plus an effective horizon condition is something I haven't seen. The paper compares Euler and RK carefully, shows the learned boundary-condition parameters (a_n,b_n)=(0.50,0.01) land close to the analytic (0.50,0.00), and clearly diagnoses why the method fails for the AdS soliton (numerical blow-up near ξ=0) and restricts to the high-temperature phase. The appendices are honest, including the note that a limited (ω,k) range can produce a range of emergent geometries.

Now the soft spots, and they are real. The training loss is a binary classification of whether the field satisfies the ingoing condition at ξ=0.1. It never compares the predicted boundary response D2/D1 to the actual dataset values. So the network is not actually trained to reproduce the measured linear response; it is trained to separate two rays in the boundary data space. The stress-test note is correct: many pairs (Ξ,Θ) can do that. There is no held-out set, no seed variation, no noise test, and no identifiability argument. The reported training loss of -1.02e-7 is impossible for a cross-entropy, so something is off in the logging or formula. Near the horizon, Θ visibly deviates, which is exactly what you'd expect if the IR part of the metric is poorly constrained. Appendix C concedes non-uniqueness.

That said, I don't think this sinks the paper. It is a proof-of-concept, and the authors are largely transparent about these caveats. The architecture is worth building on, and the central idea is plausible: with a wider range of frequencies and wavenumbers, the data should constrain the metric better. The issue is presentation and validation. The abstract and Section 5 overstate what is demonstrated: a supervised recovery on noiseless synthetic BTZ data with a coarse label, not a demonstration that a single channel 'fixes' the metric.

My recommendation: send it to peer review, but a serious referee should demand held-out validation across modes, direct supervision of the response values, error bars on the recovered metric, and a fix for the negative loss. With those, the method could be strong. As is, it's a useful contribution with an overclaimed conclusion. I'd bring it to reading group and cite it as related work.

Referee Report

4 major / 5 minor

Summary. The paper introduces an interpretable neural network whose trainable weights are the two independent metric combinations Ξ(ξ) and Θ(ξ) of a static, rotationally symmetric 3D bulk, together with a trainable boundary-condition parameter ρ_n. The architecture implements the scalar Klein-Gordon equation through repeated Runge-Kutta layers, and is trained on a synthetic dataset of boundary initial values Z_n(ξ_ini) labeled by whether they correspond to an ingoing vs. outgoing condition at the BTZ horizon. The authors report that after training, the learned (Ξ,Θ) profiles approximate the BTZ metric qualitatively, the boundary-condition parameters converge to (0.50, 0.01), and the classification accuracy reaches 1.0. They conclude in Section 5 that a single space- and time-dependent linear-response channel can fix both independent bulk metric components.

Significance. If the central claim were fully established, this would be a valuable proof-of-principle for holographic bulk reconstruction from experimentally accessible linear-response data, going beyond earlier work that used spacetime-independent nonlinear data and reconstructed only a single metric component. The paper's strengths include the Runge-Kutta layer improving numerical accuracy over the Euler method (quantified in Section 3.1.3), the use of the exact BTZ solution as a transparent benchmark, the explicit gauge-fixing argument in Appendix A, and the physics-informed regularization exploration in Appendix C. However, the current experiment does not yet validate the claimed one-to-one recovery of the metric, because the training signal is a coarse horizon-boundary classification rather than a direct comparison of boundary response data, and no held-out validation or identifiability analysis is provided. The paper is a promising proof-of-concept, but the central claim needs substantial additional support.

major comments (4)

[§3.2.4, Eqs. (22)-(24); §5] The training loss is a binary classification loss on the single number O, the magnitude of the combination ξ_fin Π_n(ξ_fin) + ρ_n Φ_n(ξ_fin) evaluated at ξ_fin = 0.1. It never compares the predicted boundary ratio D2/D1 to the values in the dataset, nor does it supervise the bulk profiles at intermediate ξ. Since the dataset labels encode only whether |C2/C1| is below 0.01 or above 1.00, the objective separates two rays in the two-dimensional boundary-data space for each (ω,k), and many pairs (Ξ,Θ) can achieve this separation without reproducing the actual linear response. Thus the central claim in Section 5 that a single space- and time-dependent linear-response channel "can fix" Ξ and Θ is not established by the presented loss function and experiment.
[§4, Fig. 8; Appendix C] Only a single training run is reported, with no held-out validation on (ω,k) modes excluded from training, no variation of random seeds, and no quantitative error bars or pointwise metric-error estimates. The visible near-horizon deviation of Θ in Fig. 8 is exactly the symptom expected when the loss is insensitive to the deep-IR part of the geometry. Moreover, Appendix C explicitly concedes that with the limited range of ω and k a range of emergent geometries may be learned. These facts undermine the identifiability assertion and require additional experiments: multiple runs with different seeds, held-out test modes, and direct comparison of the predicted response to the ground-truth response.
[§3.3, dataset construction, Eq. (27)-Eq. (28)] The dataset is generated by randomly sampling C1 and C2, labeling each sample by |C2/C1|, and then computing D1,D2 via Eq. (27); however, only the boundary value Z_n(ξ_ini) is fed to the network, while the actual response values D1,D2 are discarded from the supervised signal. As a result, the network does not learn from the source-response pair that the abstract describes as the experimental data. A direct regression loss on D2/D1, or on the full boundary asymptotics, would bring the training objective in line with the advertised experimental setup and would provide a much stronger test of metric reconstruction.
[§4, loss curve] The paper reports that the classification loss decreases to -1.02e-7, but the binary cross-entropy in Eq. (22) is non-negative for all probabilities t ∈ [0,1]. A negative reported loss indicates either a misprint, a non-standard loss implementation, or an evaluation on a different quantity. The authors should correct this and report the actual final loss value, because the quantitative accuracy of the training is one of the paper's advertised achievements.

minor comments (5)

[Abstract] "spatially and temporarily inhomogeneous" should read "spatially and temporally inhomogeneous" (or "space- and time-dependent").
[Eq. (26)-(27)] The formulas for D1_n and D2_n contain unbalanced parentheses and use symbols such as r (or r_h), γ, and ψ without explicit definition at the point of use; please rewrite these expressions with consistent notation and define all symbols.
[§3.2.2] The symbols ξ_ini and ξ_fin are used before their numerical assignments are given (0.99 and 0.10 appear later in §3.2.1 and §4); define them at first use.
[Fig. 7] The bottom panels of Fig. 7 show field values at ξ = 0.1 during training, but the caption does not identify which curves correspond to which variables or how the exact-solution values are marked; please clarify.
[Author affiliations] The affiliation line contains an obvious typesetting artifact ("Japa n" for Japan); this should be corrected in the final version.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the metric recovery is a supervised inverse-problem benchmark against the analytically known BTZ solution, not a self-referential derivation; same-author citations are motivational rather than load-bearing.

full rationale

The derivation chain is not circular. The dataset is generated from the exact BTZ solution (5): random C1 and C2 are drawn, D1 and D2 are computed from (27), initial field data Zn(ξini) are obtained from (26), and the binary label (28) encodes whether the outgoing coefficient C2 is negligible. The trainable weights are the two independent metric combinations Ξ and Θ plus the boundary-condition parameter ρn; the Runge-Kutta layer NN integrates the Klein-Gordon equation (2). Thus the training objective is a physical boundary-condition classification, not a direct copy of the true (Ξ, Θ) profiles into the weights, and the reported agreement with the BTZ metric in Fig. 8 is a nontrivial inverse-problem benchmark rather than an identity imposed by construction. The same-author citations ([1], [4], [7]) supply the SEM motivation, the sparse-NN-as-EOM idea, and an optional Einstein regularization; none is used as an unverified uniqueness theorem or as the argument that one response channel fixes the metric. Appendix A derives the gauge fixing internally, and Appendix C explicitly concedes possible multiple emergent geometries for finite (ω, k) data, which is an identifiability limitation, not a circular step. The residual near-horizon Θ deviation in Section 4 is likewise a data-coverage issue. The low score of 1 reflects only the presence of foundational same-author citations; no load-bearing step reduces to its own input.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The reconstruction rests on the standard AdS/CFT dictionary, the static rotationally symmetric metric ansatz, the gauge-fixing existence proof, the ingoing boundary condition choice, and the single-scalar-channel approximation. These are domain assumptions from the holography literature, not ad hoc inventions for this paper; the single-scalar simplification is a deliberate modeling choice compared with the two-field approach of [3]. No new physical entities are introduced.

free parameters (5)

Ξ(ξ) (89 discretized weights) = 89-point radial profile approaching BTZ (Fig. 8)
Metric combination g^{ξξ}g_{tt}; the central trainable parameter of the network, fit by backpropagation to the classification loss.
Θ(ξ) (89 discretized weights) = 89-point radial profile approaching BTZ with near-horizon deviation (Fig. 8)
Metric combination g^{ξξ}g_{θθ}; trained alongside Ξ; near-horizon values degrade due to limited ω,k_n range.
ρ_n = iω a_n + |k_n| b_n = (a_n, b_n) = (0.50, 0.01) after training
Boundary condition coefficient at the horizon; learned from data, close to BTZ value (0.50, 0.00).
Frequency/wavenumber cutoff = ω,k_n ∈ U(0,3), k_n ≤ ω
Dataset range chosen by hand; authors state larger ω,k_n are needed to resolve Θ near the horizon, so this choice limits reconstruction fidelity.
Label thresholds in Eq. (28) = 0.01 and 1.0 for |C2/C1|
Hand-chosen thresholds for positive/negative classification; affect dataset composition and training difficulty.

assumptions (6)

domain assumption AdS/CFT correspondence: boundary linear response equals bulk scalar field EOM on a fixed background.
The whole reconstruction task presumes the holographic dictionary (Section 2.1). This is the unproved conjecture at the basis of the paper.
domain assumption Bulk geometry is static and rotationally symmetric, ds^2 = g_tt(ξ)dt^2 + g_ξξ(ξ)dξ^2 + g_θθ(ξ)dθ^2.
Assumed in Section 3 for equilibrium condensed matter; reduces the metric to two functions Ξ and Θ.
domain assumption Gauge fixing √(-g)g^{ξξ} = C ξ^{-1} is always possible for asymptotically AdS static metrics.
Appendix A proves existence under regularity assumptions on the asymptotic behavior at ξ=0,1; the proof assumes a smooth bijective coordinate transformation satisfying (A.7).
domain assumption The retarded response corresponds to the ingoing boundary condition at the black hole horizon.
Used in the boundary condition layer (Eq. 16) and in the generating exact solution; standard in holography.
domain assumption A single free scalar field on a fixed background is sufficient to capture the linear response.
Footnote 2 of Section 2.1; previous work used two scalar fields, this paper uses one with higher wavenumbers; assumes the scalar channel carries enough information.
standard math Hypergeometric exact solution (6) and asymptotic expansion (26) are valid.
Standard hypergeometric solution of the Klein-Gordon equation on BTZ; used to generate the dataset and to validate solvers.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments." pith.science (2026). https://pith.science/paper/YFPY2D5B

@misc{pith2026241116052,
  author       = {Pith},
  title        = {Pith review of: Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFPY2D5B}},
  note         = {Machine review of arXiv:2411.16052}
}

read the original abstract

We introduce a novel interpretable Neural Network (NN) model designed to perform precision bulk reconstruction under the AdS/CFT correspondence. According to the correspondence, a specific condensed matter system on a ring is holographically equivalent to a gravitational system on a bulk disk, through which tabletop quantum gravity experiments may be possible as reported in arXiv:2211.13863. The purpose of this paper is to reconstruct a higher-dimensional gravity metric from the condensed matter system data via machine learning using the NN. Our machine reads spatially and temporarily inhomogeneous linear response data of the condensed matter system, and incorporates a novel layer that implements the Runge-Kutta method to achieve better numerical control. We confirm that our machine can let a higher-dimensional gravity metric be automatically emergent as its interpretable weights, using a linear response of the condensed matter system as data, through supervised machine learning. The developed method could serve as a foundation for generic bulk reconstruction, i.e., a practical solution to the AdS/CFT correspondence, and would be implemented in future tabletop quantum gravity experiments.

Figures

Figures reproduced from arXiv: 2411.16052 by the authors.

**Figure 1.** The profile of Φn(ξ) on the BTZ black hole metric is shown, comparing the exact solution with numerical results obtained using the Euler method and the Runge-Kutta method. The parameters chosen are (kn, ω) = (1.00, 1.00). The blue line represents results from the Euler method, the orange line represents results from the Runge-Kutta method, and the green dotted line represents the exact solution [PITH_FULL_IMAGE:fig… view at source ↗

**Figure 2.** The profile of Φn(ξ) on the AdS soliton metric is shown, comparing the exact solution with numerical results obtained using the Euler method and the RungeKutta method. The parameters chosen are (kn, ω) = (3.00, 3.00) and rs = 0.10. The blue line represents results from the Euler method, the orange line represents results from the Runge-Kutta method, and the green dotted line represents the exact solution. in the Ad… view at source ↗

**Figure 2.** Fig.2. We expect that this would be a generic phenomenon that happe [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Figures from the paper (6 more)

**Figure 3.** Figure 3: The model structure [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]

**Figure 4.** Figure 4: The data flow of the Runge-Kutta layer. In this study, the numerical range of ξ in the radial direction is set from 0.99 to 0.10, and the discretization is performed with an incremental width ∆ξ set to −10−2 . Therefore, the number of Runge-Kutta layers used in our mod…

**Figure 5.** Figure 5: The bulk layer. Here Z 1,2 is a component of Z and the activation f is a function of a six dimensional vector x = (x1, x2, . . . , x6) and gives the four dimensional vector (x3, x4 + x5x1 + x6x2, x5, x6). in the bulk: to determine the coefficient iω L2 2rh in (16) thou…

**Figure 6.** Figure 6: Profiles of (Ξ, Θ) are shown. The orange lines represent the pure AdS3 metric (20) which is the initial condition for the learning. The green lines represent the BTZ black hole metric (1) which is used for generating the training set and also is expected to be reproduc…

**Figure 7.** Figure 7: Top: The loss curve and the accuracy curve during the training. Bottom: evolution of the scalar field values during the training. The left (right) panel is the value of the real part of Φ (Π) at ξ = 0.1 for each epoch of the training. The boundary value of the field is…

**Figure 8.** Figure 8: Profiles of Ξ, Θ after learning. The orange line is the weights of the NN, and the green line is the ground truth, the BTZ black hole metric (1). (v) Obtain Zn(ξini) through (26) by substituting ξini for ξ. (vi) Formulate the data as {(Zn(ξini), ω2 , k2 n , tdata)}. We…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Heat and work in black hole thermodynamics via holography
hep-th 2024-12 conditional novelty 6.0 of 10

The paper derives first and second laws for composite black holes coupled through the boundary, with heat and work defined from boundary sources and bath energy flow.

Reference graph

Works this paper leans on

36 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

Spacetime-emergent ring toward tabletop quantum gravity experiments

Koji Hashimoto, Daichi Takeda, Koichiro Tanaka, and Shingo Yonezawa. Spacetime-emergent ring toward tabletop quantum gravity experiments. Physical Review Research, 5(2):023168, 2023

work page 2023
[2]

The large-n limit of superconformal ﬁeld theorie s and supergravity

Juan Maldacena. The large-n limit of superconformal ﬁeld theorie s and supergravity. International journal of theoretical physics , 38(4):1113–1133, 1999

work page 1999
[3]

Inverse problem of correlation fu nctions in holography

Bo-Wen Fan and Run-Qiu Yang. Inverse problem of correlation fu nctions in holography. JHEP, 10:228, 2024

work page 2024
[4]

Deep learning and the ads/cft correspondence

Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, and Akio Tom iya. Deep learning and the ads/cft correspondence. Physical Review D , 98(4):046019, 2018

work page 2018
[5]

Machine learning spa tial geometry from entanglement features

Yi-Zhuang You, Zhao Yang, and Xiao-Liang Qi. Machine learning spa tial geometry from entanglement features. Physical Review B , 97(4):045153, 2018. Machine-learning emergent spacetime from linear response in future tabletop quantum... 23

work page 2018
[6]

Deep learning and holographic qcd

Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, and Akio Tom iya. Deep learning and holographic qcd. Physical Review D , 98(10):106014, 2018

work page 2018
[7]

Ads/cft correspondence as a deep boltzmann machine

Koji Hashimoto. Ads/cft correspondence as a deep boltzmann machine. Physical Review D , 99(10):106017, 2019

work page 2019
[8]

Entanglement transitions from holographic random tensor networks

Romain Vasseur, Andrew C Potter, Yi-Zhuang You, and Andreas WW Ludwig. Entanglement transitions from holographic random tensor networks. Physical Review B, 100(13):134203, 2019

work page 2019

Show all 36 references

[9]

Deep learning the holographic black h ole with charge

Jing Tan and Chong-Bin Chen. Deep learning the holographic black h ole with charge. International Journal of Modern Physics D , 28(12):1950153, 2019

2019
[10]

Deep learning and ads/qcd

Tetsuya Akutagawa, Koji Hashimoto, and Takayuki Sumimoto . Deep learning and ads/qcd. Physical Review D , 102(2):026020, 2020

2020
[11]

Machin e learning holographic mapping by neural network renormalization group

Hong-Ye Hu, Shuo-Hui Li, Lei Wang, and Yi-Zhuang You. Machin e learning holographic mapping by neural network renormalization group. Physical Review Research, 2(2):023369, 2020

2020
[12]

Deep lear ning black hole metrics from shear viscosity

Yu-Kun Yan, Shao-Feng Wu, Xian-Hui Ge, and Yu Tian. Deep lear ning black hole metrics from shear viscosity. Physical Review D , 102(10):101902, 2020

2020
[13]

Neural ordin ary diﬀerential equation and holographic quantum chromodynamics

Koji Hashimoto, Hong-Ye Hu, and Yi-Zhuang You. Neural ordin ary diﬀerential equation and holographic quantum chromodynamics. Machine Learning: Science and Technology , 2(3):035011, 2021

2021
[14]

Machine learning statistical g ravity from multi-region entanglement entropy

Jonathan Lam and Yi-Zhuang You. Machine learning statistical g ravity from multi-region entanglement entropy. Physical Review Research, 3(4):043199, 2021

2021
[15]

Ads/deep-learning made easy: simple examples

Mugeon Song, Maverick SH Oh, Yongjun Ahn, and Keun-Young K im. Ads/deep-learning made easy: simple examples. Chinese Physics C , 45(7):073111, 2021

2021
[16]

Ph ysics-informed deep learning for three dimensional black holes

Emad Yaraie, Hossein Ghaﬀarnejad, and Mohammad Farsam. Ph ysics-informed deep learning for three dimensional black holes. Iranian Journal of Astrophysics and Astronomy , 10(4):335, 2023

2023
[17]

Learning the black hole metric from holographic conductivity

Kai Li, Yi Ling, Peng Liu, and Meng-He Wu. Learning the black hole metric from holographic conductivity. Physical Review D , 107(6):066021, 2023

2023
[18]

Deep learning bulk spacetime from boundary optical conductivity

Byoungjoon Ahn, Hyun-Sik Jeong, Keun-Young Kim, and Kwan Y un. Deep learning bulk spacetime from boundary optical conductivity. Journal of High Energy Physics , 2024(3):1–30, 2024

2024
[19]

Holog raphic complex potential of a quarkonium from deep learning

Mahdi Mansouri, Kazem Bitaghsir Fadafan, and Xun Chen. Holog raphic complex potential of a quarkonium from deep learning. arXiv preprint arXiv:2406.06285 , 2024

2024 arXiv
[20]

Machine learning holographic black hole f rom lattice qcd equation of state

Xun Chen and Mei Huang. Machine learning holographic black hole f rom lattice qcd equation of state. Physical Review D , 109(5):L051902, 2024

2024
[21]

Neural network modeling of heavy-quark potential from holography

Ou-Yang Luo, Xun Chen, Fu-Peng Li, Xiao-Hua Li, and Kai Zhou. Neural network modeling of heavy-quark potential from holography. arXiv preprint arXiv:2408.03784 , 2024

2024 arXiv
[22]

Flavor dependent critical endpoint fr om holographic qcd through machine learning

Xun Chen and Mei Huang. Flavor dependent critical endpoint fr om holographic qcd through machine learning. arXiv preprint arXiv:2405.06179 , 2024

2024 arXiv
[23]

Holographic reconstruction of black hole spacetime: machine learning and entanglement entropy

Byoungjoon Ahn, Hyun-Sik Jeong, Keun-Young Kim, and Kwan Yun. Holographic reconstruction of black hole spacetime: machine learning and entanglement entropy . arXiv preprint arXiv:2406.07395, 2024

2024 arXiv
[24]

Neural odes fo r holographic transport models without translation symmetry

Zhuo-Fan Gu, Yu-Kun Yan, and Shao-Feng Wu. Neural odes fo r holographic transport models without translation symmetry. arXiv preprint arXiv:2401.09946 , 2024

2024 arXiv
[25]

Gravitational duals from equations of state

Yago Bea, Raul Jimenez, David Mateos, Shuheng Liu, Pavlos Pro topapas, Pedro Taranc´ on- ´Alvarez, and Pablo Tejerina-P´ erez. Gravitational duals from equations of state. arXiv preprint arXiv:2403.14763, 2024

2024 arXiv
[26]

Anti-de Sitter space, thermal phase transitio n, and conﬁnement in gauge theories

Edward Witten. Anti-de Sitter space, thermal phase transitio n, and conﬁnement in gauge theories. Adv. Theor. Math. Phys. , 2:505–532, 1998

1998
[27]

Neural ordinary diﬀerential equations

Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary diﬀerential equations. Advances in neural information processing systems , 31, 2018

2018
[28]

Kan: Kolmogorov-arnold network s

Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halve rson, Marin Soljaˇ ci´ c, Thomas Y Hou, and Max Tegmark. Kan: Kolmogorov-arnold network s. arXiv preprint arXiv:2404.19756, 2024. Machine-learning emergent spacetime from linear response in future tabletop quan...

2024 arXiv
[29]

Attention is all you need.Advances in neural information processing systems, 30, 2017

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, /suppress Lukasz Kaiser, and Illia Polosukhin. Attention is all you need.Advances in neural information processing systems, 30, 2017

2017
[30]

End-to- end symbolic regression with transformers

Pierre-Alexandre Kamienny, St´ ephane d’Ascoli, Guillaume Lample, and Fran¸ cois Charton. End-to- end symbolic regression with transformers. Advances in Neural Information Processing Systems , 35:10269–10281, 2022

2022
[31]

Linear algebra with transformers

Fran¸ cois Charton. Linear algebra with transformers. arXiv preprint arXiv:2112.01898 , 2021

2021 arXiv
[32]

Deep learning for symbolic mathematics

Guillaume Lample and Fran¸ cois Charton. Deep learning for symbolic mathematics. arXiv preprint arXiv:1912.01412, 2019

1912 arXiv
[33]

Holography trans former

Chanyong Park, Sejin Kim, and Jung Hun Lee. Holography trans former. arXiv preprint arXiv:2311.01724, 2023

2023 arXiv
[34]

Transforming the bootstrap: Using transform ers to compute scattering amplitudes in planar n= 4 super yang-mills theory

Tianji Cai, Garrett W Merz, Fran¸ cois Charton, Niklas Nolte, Ma tthias Wilhelm, Kyle Cranmer, and Lance J Dixon. Transforming the bootstrap: Using transform ers to compute scattering amplitudes in planar n= 4 super yang-mills theory. arXiv preprint arXiv:2405.06107 , 2024

2024 arXiv
[35]

Uniﬁcation o f symmetries inside neural networks: Transformer, feedforward and neural ode

Koji Hashimoto, Yuji Hirono, and Akiyoshi Sannai. Uniﬁcation o f symmetries inside neural networks: Transformer, feedforward and neural ode. arXiv preprint arXiv:2402.02362 , 2024

2024 arXiv
[36]

Boundary Terms of the Einstein–Hilbert Action , pages 43–59

Sumanta Chakraborty. Boundary Terms of the Einstein–Hilbert Action , pages 43–59. Springer International Publishing, Cham, 2017

2017

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Spacetime-emergent ring toward tabletop quantum gravity experiments

Koji Hashimoto, Daichi Takeda, Koichiro Tanaka, and Shingo Yonezawa. Spacetime-emergent ring toward tabletop quantum gravity experiments. Physical Review Research, 5(2):023168, 2023

work page 2023

[2] [2]

The large-n limit of superconformal ﬁeld theorie s and supergravity

Juan Maldacena. The large-n limit of superconformal ﬁeld theorie s and supergravity. International journal of theoretical physics , 38(4):1113–1133, 1999

work page 1999

[3] [3]

Inverse problem of correlation fu nctions in holography

Bo-Wen Fan and Run-Qiu Yang. Inverse problem of correlation fu nctions in holography. JHEP, 10:228, 2024

work page 2024

[4] [4]

Deep learning and the ads/cft correspondence

Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, and Akio Tom iya. Deep learning and the ads/cft correspondence. Physical Review D , 98(4):046019, 2018

work page 2018

[5] [5]

Machine learning spa tial geometry from entanglement features

Yi-Zhuang You, Zhao Yang, and Xiao-Liang Qi. Machine learning spa tial geometry from entanglement features. Physical Review B , 97(4):045153, 2018. Machine-learning emergent spacetime from linear response in future tabletop quantum... 23

work page 2018

[6] [6]

Deep learning and holographic qcd

Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, and Akio Tom iya. Deep learning and holographic qcd. Physical Review D , 98(10):106014, 2018

work page 2018

[7] [7]

Ads/cft correspondence as a deep boltzmann machine

Koji Hashimoto. Ads/cft correspondence as a deep boltzmann machine. Physical Review D , 99(10):106017, 2019

work page 2019

[8] [8]

Entanglement transitions from holographic random tensor networks

Romain Vasseur, Andrew C Potter, Yi-Zhuang You, and Andreas WW Ludwig. Entanglement transitions from holographic random tensor networks. Physical Review B, 100(13):134203, 2019

work page 2019