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A neural network reconstructs the holographic QCD background from unflavored meson masses, then uses it to predict pion masses.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 15:30 UTC pith:3ORNYHBJ

load-bearing objection A useful, reproducible extension of NN-holographic fitting, but the pion 'prediction' is not independent and the abstract disagrees with the main text on headline numbers. the 4 major comments →

arxiv 2512.16450 v3 pith:3ORNYHBJ submitted 2025-12-18 hep-ph cond-mat.dis-nnhep-th

Learning holographic QCD with unflavored meson spectra

Mathew Thomas Arun , Ritik Pal This is my paper
classification hep-ph cond-mat.dis-nnhep-th
keywords holographic QCDAdS/QCDneural networkmeson spectrumdilaton profilechiral symmetry breakinginverse problemradial excitations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that a data-driven neural network can solve the inverse problem of holographic QCD: given the experimentally measured masses of the rho, a1, a2, and f0 mesons and their excitations, the network reconstructs the warp factor, the dilaton profile, and the chiral-symmetry-breaking scalar potential without assuming their functional forms. Trained on those four towers, the model matches the input masses to within a few percent and then predicts the pion ground state and excited states, with good agreement for the ground and second excited states. The authors claim the learned dilaton is positive everywhere and grows between linearly and quadratically in the infrared, and that the scalar potential requires both cubic and quartic terms, with k1 near -7.8 and k2 near 17. A sympathetic reader would care because this turns spectroscopy into a direct probe of the extra-dimensional background: instead of choosing a profile by hand, the mass spectrum itself selects it.

Core claim

The central claim is that one neural-network reconstruction of the geometry—the warp factor A(z), the scalar VEV v(z), and the dilaton profile phi(z)—together with a bulk potential V(X)=k1 X^3 + k2 X^4 reproduces the full listed spectra of the rho, a1, a2, and f0 mesons and, without retraining, predicts pion masses. The reconstruction is constrained only by UV asymptotic forms and by penalties enforcing positive v'(z), confining IR potentials, and a negative warp factor in the IR. From the learned v(z) and the GMOR relation the model extracts a chiral condensate of about (0.300 GeV)^3 and a quark mass of about 3 MeV. The paper reports the pion ground state at 0.161 +/- 0.057 GeV (experiment

What carries the argument

The engine is the Schrodinger-like eigenvalue problem for each meson channel, discretized on a lattice of z points with Dirichlet boundary conditions. The effective potentials V_rho, V_a1, V_a2, and V_f0 are built from the learned warp factor, dilaton, and scalar VEV; the discretized second derivative turns each potential into a real symmetric tridiagonal matrix whose eigenvalues are the squared meson masses. Because the eigenvalues are differentiable through the Hellmann-Feynman formula dλ/dw = q^T (dH/dw) q, gradient descent can update the neural networks and the four scalar parameters (k1, k2, L, theta) directly. The pion sector is handled separately as a generalized eigenvalue problem fr

Load-bearing premise

The claim that the pion spectrum is a successful prediction rests on the pion channel not leaking into training; the experimental pion mass sets the quark mass through GMOR, pion errors appear in the initial-guess objective, and the reported best-fit numbers come from the run with the lowest pion loss.

What would settle it

Train the same architecture with the pion channel completely excluded: fix the quark mass and chiral condensate from lattice QCD, remove pion terms from the hyperparameter objective, and select runs by training loss only; if the predicted pion masses then shift away from experiment, the claimed independent prediction was an artifact of target leakage.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the reconstruction is correct, the meson spectrum alone fixes the warp factor, dilaton, and chiral condensate, so bottom-up holographic QCD models no longer need hand-picked profile functions.
  • The positive, super-linear dilaton reconciles confinement with the null energy condition, avoiding the continuum spectrum that a purely linear dilaton would produce for glueballs.
  • The extracted quark mass and condensate from GMOR are in the expected ballpark, so the method can act as a spectroscopy-based determination of low-energy constants.
  • The pion prediction, especially the ground and second excited states, is evidence that the learned background transfers to a channel not used in training.
  • The same pipeline can be extended to other observables, such as decay constants or additional channels, by adding loss terms, which the paper identifies as a flexible next step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A cleaner out-of-sample test would exclude the pion mass entirely from parameter fixing: set the quark mass from lattice input and drop pion terms from both the hyperparameter search and the run selection; the current pipeline uses the experimental pion mass in the GMOR relation and picks the run with the lowest pion loss, so the published pion agreement may overstate generalization.
  • The paper itself notes that adding the cubic term significantly reduced the loss and that this could indicate insufficient hyperparameter tuning in the quartic-only case; a broader search over quartic-only models would settle whether k1 is physically required or is an artifact of the training procedure.
  • The overshoot of the first excited pion suggests the learned infrared potential is not yet accurate at intermediate energies; including pion decay constants or additional radial states in training could sharpen the reconstructed geometry.
  • If the method transfers to other sectors, the same trained background could be used to predict glueball masses or decay constants, quantities not touched in the paper.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 2 minor

Summary. The paper proposes a neural-network framework to solve the inverse problem of holographic QCD: from the masses of unflavored mesons (ρ, a1, a2, f0 and excitations), it trains networks for the warp factor A(z), the scalar VEV v(z), the dilaton φ(z), and the scalar-potential coefficients k1, k2. The mass spectra are computed by finite-difference discretization of the Schrödinger-like equations into a tridiagonal eigenvalue problem, and gradients are propagated through the eigenvalues via the Hellmann–Feynman theorem. The trained model is then used to compute the pion spectrum, which the paper reports as a successful prediction. The central claim is that the learned background geometry and potentials reproduce the training masses and independently predict the pion masses.

Significance. If the claims were fully substantiated, the paper would offer a flexible, assumption-light method for reconstructing holographic QCD backgrounds from hadron spectra. The finite-difference eigenvalue pipeline with differentiable eigenvalue solvers is a useful technical contribution, and the public release of code and trained models is a clear strength. However, the validation of the main claim is currently undermined by target leakage: the experimental pion mass enters through the GMOR relation in setting the VEV boundary condition, the TPE hyperparameter objective explicitly includes pion errors, and the best pion fit is selected using the lowest pion loss. With these channels, the reported pion spectrum is not an independent prediction. The abstract also contradicts the body on the central numerical results. The framework may still be salvageable, but the paper as written does not support its generalization claim.

major comments (4)
  1. [§3.1, Eq. (7), Eq. (23)] The quark mass m_q is computed from the GMOR relation (Eq. 7) using the experimental pion mass, and m_q then fixes α and β in the VEV ansatz (Eqs. 5–6, 23). Since v(z) controls all effective potentials, the experimental m_π is an input to the very geometry that is supposed to predict the pion spectrum. This makes the pion ground state a postdiction rather than a prediction. The authors should redo the analysis with m_q fixed from an independent source (e.g., from the current-quark mass and the trained Σ) and report the resulting pion masses.
  2. [Appendix B, Eq. (50); Table 5] The TPE objective O(k1,k2) used to choose the initial values of k1 and k2 explicitly sums pion errors (Eq. 50). Since k1,k2 strongly influence the potentials and the final trained values depend on their initialization, the pion spectrum participates in model selection. In addition, Table 4 and the surrounding text select the 'best fit' pion predictions as the run with the lowest L(π)_mass, i.e., the test metric is used for model selection. Both channels leak target information into the reported prediction and invalidate the claim of independent validation.
  3. [Abstract vs. §1/Table 2] The abstract reports k1 ∼ −4 and k2 ∼ 9, and states that the dilaton's IR behavior is 'much steeper than its quadratic form.' The main text reports k1 = −7.77 ± 1.05, k2 = 17.07 ± 2.75 (Table 2) and a dilaton profile 'in-between linear and quadratic' (§1, §5, Fig. 2). These are incompatible statements of the core results. The abstract must be corrected to match the actual findings, and the provenance of the numbers should be clarified.
  4. [Eq. (2) vs. Eq. (26)] The action in Eq. (2) defines V(X) = (4/3)k1 X^3 + 2k2 X^4, whose derivative is 4k1 X^2 + 8k2 X^3. However, the equations of motion actually used, Eqs. (4) and (26), treat V_k(v) = k1 v^2 + k2 v^3 as the derivative, which corresponds to V(X) = (k1/3)X^3 + (k2/4)X^4. The reported k1,k2 therefore do not correspond to the advertised scalar potential. This inconsistency should be resolved and stated clearly, as it affects the interpretation of the learned potential.
minor comments (2)
  1. [Throughout] There are several typos, including 'dilton' in the abstract and Introduction, and 'k, and L' in §3.2 (should be k2). Please proofread.
  2. [Fig. 2 caption] The left panel of Fig. 2 is said to show dφ/dz for the best run, while the right panel shows the mean φ(z). Clarify in the caption which average and error band are used, and why the left panel is not shown with the mean/standard deviation like the other figures.

Circularity Check

3 steps flagged

Pion 'prediction' leaks target data via GMOR input, TPE objective, and run selection; generalization claim is not independent.

specific steps
  1. self definitional [Sec. 3.1 (Model Architecture), after Eq. (23); Eqs. (6), (7), (37)]
    "The quark mass m_q is then computed using the GMOR relation given in Eq. 7 and this is, in turn, used to compute the α and β values from Eq. 6. [Eq. 7: f_π^2 m_π^2 = 2 m_q Σ]"

    The experimental pion mass m_π is the quantity the paper claims to predict (Table 4). Here m_π fixes m_q through GMOR, and m_q fixes α in v(z)=αz+βz^3+... via Eqs. (6) and (23). The same v(z) enters the coupled pion equations (Eq. 13) through ω(z) and C(z), so the predicted pion spectrum is not an independent test: target data are used to construct the background that produces the prediction.

  2. fitted input called prediction [Appendix B, Eq. (50); Sec. 3.1 initial-guess paragraph]
    "The objective function for the optimization is defined as O(k1, k2) = Σ_{μ∈{ρ,a1,a2,f0,π}} (1/N_μ) Σ_{n=1}^{N_μ} E_{μ,n}. ... The best values of k1 and k2 obtained from the optimization are k1 = −7.4 and k2 = 12.2. These values are used as the initial guess for k1 and k2 in the subsequent training runs."

    The TPE initial guess for k1 and k2—the two scalar-potential coefficients that control all the meson potentials—is selected by minimizing an objective that explicitly includes pion errors. Although the later gradient training of L_mass omits π, the reported model starts from a point chosen using the target data, so the pion 'prediction' depends on target values through the initialization of the fit.

  3. other [Sec. 4, Table 4 and Table 5]
    "Here, the best fit values are obtained for the run with the lowest value of L^{(π)}_{mass}. The Loss corresponding to the π meson masses for each run is given in Table 5."

    The paper reports as its pion prediction the run selected by the smallest pion loss L_π. Selecting the model on the target metric means the quoted 'best fit' pion masses are not an unbiased out-of-sample prediction; target data were used for model selection. The paper itself notes Run 6 'yielded the lowest π mass loss ... despite not being trained for it', confirming that selection, not trained generalization, drives the best-fit numbers.

full rationale

The training of the rho, a1, a2, and f0 towers is a genuine inverse-problem fit: the mass loss in Eq. (32) contains only those four channels and the network differentiates through finite-difference eigenvalues. That part is not circular. The circularity is in the claimed out-of-sample validation. First, the experimental pion mass is inverted via GMOR to set m_q, which fixes the UV coefficients of v(z); the same v(z) appears in the pion equations, so the pion ground state is an input to the model rather than a pure prediction. Second, the TPE initialization of k1, k2 in Eq. (50) explicitly minimizes pion errors, biasing the fit basin toward target values. Third, the reported best-fit pion masses are chosen by lowest L_π among ten runs, i.e. the test metric is used as a model-selection criterion. These three channels jointly undermine the central claim that the learned geometry 'predicts' the pion spectrum. I do not set the score higher (8–10) because the four training channels are fit honestly and the pion eigenvalues are not literally set equal to the experimental values—indeed the first excited pion is overpredicted by ~28%—but the central generalization test is not independent. The abstract/body discrepancy for k1,k2 and for the dilaton IR behavior is a separate correctness risk, not an additional circular step.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central claim rests on the bottom-up AdS/QCD model as a given, on the identification of bulk eigenvalues with meson masses, and on the GMOR relation that ties the pion mass into the input. The NN weights and the parameters k1,k2,L,theta absorb the fitting freedom; the method does not introduce new entities.

free parameters (5)
  • k1 (cubic scalar-potential coefficient) = -7.77 +/- 1.05 (run range -9.85 to -6.30)
    Trainable parameter optimized to fit rho/a1/a2/f0 masses (Eq.31); reported as a prediction though it is a fit.
  • k2 (quartic scalar-potential coefficient) = 17.07 +/- 2.75 (run range 13.55 to 22.52)
    Same as k1: trainable parameter in the mass loss.
  • L (AdS radius) = 1.71 +/- 0.26 GeV^-1
    Derived from trainable l via L=log(1+l^2), fitted to the meson masses.
  • theta (condensate parameter) = 2.41 +/- 1.32; Sigma=(0.300 +/- 0.013 GeV)^3, m_q=0.0031 +/- 0.0005 GeV
    Determines the chiral condensate via Eq.24; m_q is computed from GMOR.
  • NN weights of A(z) and v(z) networks = two 4x50-layer MLPs, thousands of weights
    All weights trained to minimize the mass loss; these functions are the reconstruction output.
axioms (5)
  • domain assumption Bottom-up 5D AdS/QCD action with m_X^2=-3 and V(X)=4/3 k1 X^3 + 2 k2 X^4 (Eq.2)
    Defines the model space; taken as given, not derived from string theory.
  • domain assumption Asymptotic AdS boundary conditions: A(z)->-log z and v(z)->alpha z + beta z^3 (Eq.5)
    Input to the NN parameterization in Eq.23.
  • domain assumption GMOR relation f_pi^2 m_pi^2 = 2 m_q Sigma (Eq.7) used to fix m_q
    Load-bearing for the pion prediction; leaks target data into the training setup.
  • domain assumption Meson masses are eigenvalues of the Schrodinger-like equations (Eqs.9,15,18,21)
    Standard holographic dictionary used throughout the paper.
  • domain assumption v(z) non-decreasing and confining IR potentials enforced via penalties (Eqs.33-35)
    Needed to stabilize the inverse problem; restricts the solution space.

pith-pipeline@v1.3.0-alltime-deepseek · 16806 in / 13474 out tokens · 123555 ms · 2026-08-03T15:30:49.196250+00:00 · methodology

0 comments
read the original abstract

We develop a data-driven neural network framework to reconstruct the five-dimensional background geometry, the dilaton potential, and the chiral-symmetry-breaking scalar potential of holographic QCD from hadron mass spectra. Framed as an inverse problem, the model is trained using a discretized form of the Schr\"odinger-like equation, which resembles a linear moose in ``deconstructed" 5 dimensions with Dirichlet boundary conditions, in contrast to the AdS/DL with ``emergent" space-time. Using the masses of the unflavored mesons $\rho$, $a_1$, $a_2$, and $f_0$ and their excitations as training data, the model learns confining effective potentials and computes a dilaton profile that satisfies the null energy condition. The network predicts that the dilaton's IR behavior will be much steeper than its quadratic form. Moreover, the symmetry-breaking bulk potential of the scalar field, $V(X) \sim k_1 X^3+k_2 X^4$, was computed, and the parameters $k_1$ and $k_2$ predicted to be $\sim -4$ and $\sim 9$ respectively. The deep-learned parameters, metric, and the dilaton profile were then used to predict the pion mass and its spectrum with good accuracy. A Python code, along with the trained models, is provided to facilitate further studies\footnote{Available at Github, https://github.com/rp-winter/NN-AdS-QCD

Figures

Figures reproduced from arXiv: 2512.16450 by Mathew Thomas Arun, Ritik Pal.

Figure 1
Figure 1. Figure 1: The learned functions v(z) (left) and A(z) (right) after training. The shaded regions represent the standard deviation over different runs, and the solid lines represent the mean value. 0 2 4 6 8 10 z 0 2 4 6 8 10 12 d /dz NN: Lowest ( , a1, a2, f0) mass 0 2 4 6 8 10 z 0 10 20 30 40 50 60 (z) 4.88z 0.64z 2 NN: Mean [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The plot of dϕ/dz (left) and ϕ(z) (right). The solid line represents the best fit corresponding to the minimum value of L (ρ,a1,a2,f0) mass . Unlike Figs. 1 and 2, where the spread was symmetric and shown as mean ± standard deviation, the dϕ/dz here exhibit a skewed variation, which is why we only plot the dϕ/dz corresponding to the best run. However, the ϕ(z) (right panel) obtained shows a symmetric sprea… view at source ↗
Figure 3
Figure 3. Figure 3: Effective potentials for the ρ, a1, a2, and f0 mesons. The solid line represents the best fit corresponding to the minimum value of L (ρ,a1,a2,f0) mass . parameters and functions. This time, we obtain a matrix for a generalized eigenvalue equation (see Appendix A for more details), which we solve using SciPy’s built-in function scipy.linalg.eigvals. The predicted masses eigenvalues mn obtained (after conve… view at source ↗
Figure 4
Figure 4. Figure 4: Mass spectra of the ρ, f0, a1 and a2 mesons [47]. The predicted masses correspond to the mean, and the error bars show the standard deviation across multiple runs. 1 2 3 n 0.25 0.50 0.75 1.00 1.25 1.50 1.75 M ass of m eso n (G e V) Predicted Experimental Individual Runs [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mass spectrum of the predicted π meson as shown in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The loss curve for a representative training run. [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

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Reference graph

Works this paper leans on

51 extracted references · 25 linked inside Pith · cited by 2 Pith papers

  1. [1]

    J. M. Maldacena,The LargeNlimit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231–252, [hep-th/9711200]

    Pith/arXiv arXiv 1998

  2. [2]

    S. S. Gubser, I. R. Klebanov and A. M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105–114, [hep-th/9802109]. 18

    Pith/arXiv arXiv 1998

  3. [3]

    Witten,Anti de Sitter space and holography,Adv

    E. Witten,Anti de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253–291, [hep-th/9802150]

    Pith/arXiv arXiv 1998

  4. [4]

    Sakai and S

    T. Sakai and S. Sugimoto,Low energy hadron physics in holographic QCD,Prog. Theor. Phys. 113(2005) 843–882, [hep-th/0412141]

    Pith/arXiv arXiv 2005

  5. [5]

    Sakai and S

    T. Sakai and S. Sugimoto,More on a holographic dual of QCD,Prog. Theor. Phys.114(2005) 1083–1118, [hep-th/0507073]

    Pith/arXiv arXiv 2005

  6. [6]

    Kruczenski, D

    M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters,Towards a holographic dual of large N(c) QCD,JHEP05(2004) 041, [hep-th/0311270]

    Pith/arXiv arXiv 2004

  7. [7]

    Hashimoto, T

    K. Hashimoto, T. Hirayama, F.-L. Lin and H.-U. Yee,Quark Mass Deformation of Holographic Massless QCD,JHEP07(2008) 089, [0803.4192]

    Pith/arXiv arXiv 2008

  8. [8]

    Holdom and M

    B. Holdom and M. E. Peskin,Raising the Axion Mass,Nucl. Phys. B208(1982) 397–412

    1982

  9. [9]

    Holdom,Strong QCD at High-energies and a Heavy Axion,Phys

    B. Holdom,Strong QCD at High-energies and a Heavy Axion,Phys. Lett. B154(1985) 316. [Erratum: Phys.Lett.B 156, 452 (1985)]

    1985

  10. [10]

    Erlich, E

    J. Erlich, E. Katz, D. T. Son and M. A. Stephanov,Qcd and a holographic model of hadrons, Physical Review Letters95(Dec., 2005)

    2005

  11. [11]

    J. Hirn, N. Rius and V. Sanz,Geometric approach to condensates in holographic QCD,Phys. Rev. D73(2006) 085005, [hep-ph/0512240]

    Pith/arXiv arXiv 2006

  12. [12]

    Karch, E

    A. Karch, E. Katz, D. T. Son and M. A. Stephanov,Linear confinement and AdS/QCD,Phys. Rev. D74(2006) 015005, [hep-ph/0602229]

    Pith/arXiv arXiv 2006

  13. [13]

    Csaki and M

    C. Csaki and M. Reece,Toward a systematic holographic QCD: A Braneless approach,JHEP 05(2007) 062, [hep-ph/0608266]

    Pith/arXiv arXiv 2007

  14. [14]

    Falkowski and M

    A. Falkowski and M. Perez-Victoria,Holography, pade approximants and deconstruction,JHEP 02(2007) 086, [hep-ph/0610326]

    Pith/arXiv arXiv 2007

  15. [15]

    J. P. Shock, F. Wu, Y.-L. Wu and Z.-F. Xie,AdS/QCD Phenomenological Models from a Back-Reacted Geometry,JHEP03(2007) 064, [hep-ph/0611227]

    Pith/arXiv arXiv 2007

  16. [16]

    Karch, E

    A. Karch, E. Katz, D. T. Son and M. A. Stephanov,On the sign of the dilaton in the soft wall models,JHEP04(2011) 066, [1012.4813]

    Pith/arXiv arXiv 2011

  17. [17]

    Gherghetta, J

    T. Gherghetta, J. I. Kapusta and T. M. Kelley,Chiral symmetry breaking in the soft-wall ads/qcd model,Physical Review D79(Apr., 2009)

    2009

  18. [18]

    Zhang,Mesons and nucleons in soft-wall ads/qcd,Physical Review D82(Nov., 2010)

    P. Zhang,Mesons and nucleons in soft-wall ads/qcd,Physical Review D82(Nov., 2010)

    2010

  19. [19]

    Sui, Y.-L

    Y.-Q. Sui, Y.-L. Wu, Z.-F. Xie and Y.-B. Yang,Prediction for the mass spectra of resonance mesons in the soft-wall ads/qcd model with a modified 5d metric,Phys. Rev. D81(Jan, 2010) 014024

    2010

  20. [20]

    Xie, J.-G

    H. Xie, J.-G. Liu and L. Wang,Automatic differentiation of dominant eigensolver and its applications in quantum physics,Phys. Rev. B101(Jun, 2020) 245139

    2020

  21. [21]

    de Paula, T

    W. de Paula, T. Frederico, H. Forkel and M. Beyer,Dynamical holographic qcd with area-law confinement and linear regge trajectories,Phys. Rev. D79(Apr, 2009) 075019. 19

    2009

  22. [22]

    Ballon-Bayona, T

    A. Ballon-Bayona, T. Frederico, L. A. H. Mamani and W. de Paula,Dynamical holographic qcd model for spontaneous chiral symmetry breaking and confinement,Phys. Rev. D108(Nov,

  23. [23]

    L. A. H. Mamani,Conformal symmetry breaking in holographic qcd,Phys. Rev. D100(Nov,

  24. [24]

    Hornik, M

    K. Hornik, M. Stinchcombe and H. White,Multilayer feedforward networks are universal approximators,Neural Networks2(1989) 359–366

    1989

  25. [25]

    Hashimoto,AdS/CFT correspondence as a deep Boltzmann machine,Phys

    K. Hashimoto,AdS/CFT correspondence as a deep Boltzmann machine,Phys. Rev. D99 (2019) 106017, [1903.04951]

    Pith/arXiv arXiv 2019

  26. [26]

    Hashimoto, S

    K. Hashimoto, S. Sugishita, A. Tanaka and A. Tomiya,Deep Learning and Holographic QCD, Phys. Rev. D98(2018) 106014, [1809.10536]

    Pith/arXiv arXiv 2018

  27. [27]

    Akutagawa, K

    T. Akutagawa, K. Hashimoto and T. Sumimoto,Deep learning and ads/qcd,Physical Review D 102(July, 2020)

    2020

  28. [28]

    Hashimoto, K

    K. Hashimoto, K. Ohashi and T. Sumimoto,Deriving the dilaton potential in improved holographic QCD from the meson spectrum,Phys. Rev. D105(2022) 106008, [2108.08091]

    Pith/arXiv arXiv 2022

  29. [29]

    Pal,Github,https://github.com/rp-winter/NN-AdS-QCD(2025)

    R. Pal,Github,https://github.com/rp-winter/NN-AdS-QCD(2025)

    2025

  30. [30]

    Arkani-Hamed, A

    N. Arkani-Hamed, A. G. Cohen and H. Georgi,(De)constructing dimensions,Phys. Rev. Lett. 86(2001) 4757–4761, [hep-th/0104005]

    Pith/arXiv arXiv 2001

  31. [31]

    Cheng, C

    H.-C. Cheng, C. T. Hill and J. Wang,Dynamical Electroweak Breaking and Latticized Extra Dimensions,Phys. Rev. D64(2001) 095003, [hep-ph/0105323]

    Pith/arXiv arXiv 2001

  32. [32]

    H. Abe, T. Kobayashi, N. Maru and K. Yoshioka,Field localization in warped gauge theories, Phys. Rev. D67(2003) 045019, [hep-ph/0205344]

    Pith/arXiv arXiv 2003

  33. [33]

    Randall, Y

    L. Randall, Y. Shadmi and N. Weiner,Deconstruction and gauge theories in AdS(5),JHEP01 (2003) 055, [hep-th/0208120]

    Pith/arXiv arXiv 2003

  34. [34]

    Falkowski and H

    A. Falkowski and H. D. Kim,Running of gauge couplings in AdS(5) via deconstruction,JHEP 08(2002) 052, [hep-ph/0208058]

    Pith/arXiv arXiv 2002

  35. [35]

    D. T. Son and M. A. Stephanov,QCD and dimensional deconstruction,Phys. Rev. D69(2004) 065020, [hep-ph/0304182]

    Pith/arXiv arXiv 2004

  36. [36]

    de Blas, A

    J. de Blas, A. Falkowski, M. Perez-Victoria and S. Pokorski,Tools for deconstructing gauge theories in AdS(5),JHEP08(2006) 061, [hep-th/0605150]

    Pith/arXiv arXiv 2006

  37. [37]

    Erlich, G

    J. Erlich, G. D. Kribs and I. Low,Emerging holography,Phys. Rev. D73(2006) 096001, [hep-th/0602110]

    Pith/arXiv arXiv 2006

  38. [38]

    Nakai,Deconstruction, Holography and Emergent Supersymmetry,JHEP03(2015) 101, [1412.3486]

    Y. Nakai,Deconstruction, Holography and Emergent Supersymmetry,JHEP03(2015) 101, [1412.3486]

    Pith/arXiv arXiv 2015

  39. [39]

    Kiritsis and F

    E. Kiritsis and F. Nitti,On massless 4d gravitons from asymptotically ads5 space–times,Nuclear Physics B772(2007) 67–102

    2007

  40. [40]

    Chelabi, Z

    K. Chelabi, Z. Fang, M. Huang, D. Li and Y.-L. Wu,Chiral phase transition in the soft-wall model of ads/qcd,Journal of High Energy Physics2016(Apr., 2016) 1–30. 20

    2016

  41. [41]

    G¨ ursoy and E

    U. G¨ ursoy and E. Kiritsis,Exploring improved holographic theories for qcd: part i,Journal of High Energy Physics2008(feb, 2008) 032

    2008

  42. [42]

    Cherman, T

    A. Cherman, T. D. Cohen and E. S. Werbos,Chiral condensate in holographic models of qcd, Phys. Rev. C79(Apr, 2009) 045203

    2009

  43. [43]

    T. M. Kelley, S. P. Bartz and J. I. Kapusta,Pseudoscalar mass spectrum in a soft-wall model of ads/qcd,Phys. Rev. D83(Jan, 2011) 016002

    2011

  44. [44]

    Ballon-Bayona and L

    A. Ballon-Bayona and L. A. H. Mamani,Nonlinear realization of chiral symmetry breaking in holographic soft wall models,Phys. Rev. D102(Jul, 2020) 026013

    2020

  45. [45]

    Ozaki, Y

    Y. Ozaki, Y. Tanigaki, S. Watanabe and M. Onishi,Multiobjective tree-structured parzen estimator for computationally expensive optimization problems, inProceedings of the 2020 Genetic and Evolutionary Computation Conference, GECCO ’20, (New York, NY, USA), p. 533–541, Association for Computing Machinery, 2020. DOI

    2020

  46. [46]

    Akiba, S

    T. Akiba, S. Sano, T. Yanase, T. Ohta and M. Koyama,Optuna: A next-generation hyperparameter optimization framework, inProceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2019. [47]Particle Data Groupcollaboration, S. Navas et al.,Review of particle physics,Phys. Rev. D 110(2024) 030001

    2019

  47. [48]

    Hashimoto, S

    K. Hashimoto, S. Sugishita, A. Tanaka and A. Tomiya,Deep learning and theAdS/CFT correspondence,Phys. Rev. D98(Aug, 2018) 046019

    2018

  48. [49]

    R. P. Feynman,Forces in molecules,Phys. Rev.56(Aug, 1939) 340–343

    1939

  49. [50]

    Ramachandran, B

    P. Ramachandran, B. Zoph and Q. V. Le,Searching for activation functions, 2017

    2017

  50. [51]

    K. He, X. Zhang, S. Ren and J. Sun,Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, 2015

    2015

  51. [52]

    D. P. Kingma and J. Ba,Adam: A method for stochastic optimization, 2017. 21

    2017