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arxiv: 2604.02906 · v1 · submitted 2026-04-03 · ✦ hep-ph · hep-th

Recognition: no theorem link

Probing Proton Structure via Physics-Guided Neural Networks in Holographic QCD

Wei Kou , Xurong Chen
Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:41 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords holographic QCDdeep inelastic scatteringphysics-guided neural networkproton structure functionPomeron exchangeAdS5 Dirac equation
0
0 comments X

The pith

A neural network embedding the AdS5 Dirac equation and string diffusion kernel fits SLAC proton data and extracts the resonance-to-Pomeron transition at x ≈ 0.19.

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

The paper presents a Physics-Guided Neural Network that directly incorporates the five-dimensional AdS5 Dirac equation and the string diffusion kernel into its computational graph. This embedding forces the network to respect the physical proton mass while training on high-precision SLAC deep inelastic scattering measurements. The resulting model achieves a global fit quality of χ²/d.o.f. ≈ 0.91 without using preset functional forms for the structure function F2. Instead, the optimization itself identifies the kinematic point where s-channel bulk fermion contributions give way to t-channel holographic Pomeron exchange and recovers a Pomeron intercept near 1.0786.

Core claim

Embedding the AdS5 Dirac equation and string diffusion kernel into the neural network computational graph constrains the model to the physical proton mass and produces a data-driven description of F2. The network automatically locates the transition between hadronic resonance excitations and diffractive Pomeron background near x ≈ 0.19, recovers the Pomeron intercept α0 ≈ 1.0786, and generates higher-twist scale breaking through the evolution of resonance mass spectra.

What carries the argument

Physics-Guided Neural Network (PGNN) that embeds the five-dimensional AdS5 Dirac equation and the string diffusion kernel directly into the computational graph.

If this is right

  • The network identifies a kinematic crossover from resonance to Pomeron regime near x ≈ 0.19 without prior parametrization.
  • Optimization recovers the Pomeron intercept α0 ≈ 1.0786 as an emergent output.
  • Higher-twist scale-breaking effects arise naturally from resonance mass spectra evolution.
  • The same embedding approach supplies an interpretable description of non-perturbative and transition regimes in QCD.

Where Pith is reading between the lines

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

  • The method could be applied to other structure functions or to nuclei once corresponding holographic kernels are available.
  • If the recovered intercept proves stable under changes in network depth, it would support the holographic Pomeron picture as a data-driven result rather than an input assumption.
  • Combining the PGNN with lattice QCD inputs might test whether the extracted transition point x ≈ 0.19 persists across different non-perturbative regulators.

Load-bearing premise

Directly embedding the five-dimensional AdS5 Dirac equation and string diffusion kernel into the neural network graph constrains the model to the physical proton mass without uncontrolled biases from network architecture or training.

What would settle it

Independent data sets yielding a significantly worse global χ²/d.o.f. or a Pomeron intercept far from 1.0786 would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.02906 by Wei Kou, Xurong Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. The holographic dual-channel physical picture for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic architecture of the PGNN proposed in this work. The kinematic inputs ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The neural-network-extracted mechanism weight [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The physical constraints and dynamical outputs of the PGNN holographic parameters. (a) The proton mass manifold [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The training dynamics and parameter convergence of the PGNN framework. (Left) The evolution of the total loss [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Describing the proton structure function $F_2$ in the non-perturbative and transition regimes of quantum chromodynamics (QCD) remains a significant theoretical challenge. In this work, we introduce a Physics-Guided Neural Network (PGNN) that integrates Holographic QCD with deep learning. By embedding the five-dimensional $\text{AdS}_5$ Dirac equation and the string diffusion kernel directly into the computational graph, the network is strictly constrained to the physical proton mass ($M_p \equiv 0.938 \text{ GeV}$). Applying this framework to high-precision SLAC deep inelastic scattering data yields a global fit of $\chi^2/\text{d.o.f.} \simeq 0.91$. Rather than relying on predetermined empirical forms, the network dynamically extracts the transition between the $s$-channel bulk fermion mechanism (hadronic resonance excitations) and the $t$-channel holographic Pomeron exchange (diffractive background), identifying a kinematic crossover near $x \approx 0.19$. Furthermore, the optimization naturally recovers a Pomeron intercept of $\alpha_0 \approx 1.0786$ and generates higher-twist scale-breaking effects through the evolution of resonance mass spectra. This demonstrates that embedding analytical differential equations into neural networks provides an interpretable, data-driven approach for phenomenological studies of strongly coupled systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a Physics-Guided Neural Network (PGNN) that embeds the five-dimensional AdS5 Dirac equation and string diffusion kernel directly into the computational graph to enforce the physical proton mass Mp = 0.938 GeV while fitting the proton structure function F2 to high-precision SLAC deep inelastic scattering data. It reports a global fit with χ²/d.o.f. ≃ 0.91, claims the network dynamically extracts the transition between s-channel bulk fermion resonances and t-channel holographic Pomeron exchange at x ≈ 0.19, and states that optimization naturally recovers a Pomeron intercept α0 ≈ 1.0786 along with higher-twist effects from resonance mass spectra evolution.

Significance. If the embedding enforces the physical constraints without uncontrolled biases, the work demonstrates a promising data-driven yet interpretable framework for modeling non-perturbative QCD regimes by combining holographic models with neural networks. The reported fit quality and dynamical extraction of the kinematic crossover would provide a concrete phenomenological tool for studying the resonance-to-Pomeron transition, with potential extensions to other strongly coupled systems.

major comments (2)
  1. [Abstract] Abstract: The assertion that embedding the AdS5 Dirac equation 'strictly constrains' the network to Mp ≡ 0.938 GeV requires explicit demonstration that this is a hard constraint (e.g., via architectural enforcement or infinite penalty weight) rather than a finite-weighted residual term in the loss function. Without reporting the residual norm on the Dirac operator or sensitivity tests to the embedding hyperparameter, the recovered α0 ≈ 1.0786 and x ≈ 0.19 crossover may partly reflect training dynamics or network biases instead of pure holographic QCD.
  2. [Abstract] Abstract and results section: The claim that the optimization 'naturally recovers' the Pomeron intercept α0 ≈ 1.0786 is load-bearing for the interpretation of unbiased extraction, yet α0 is a standard free parameter in holographic Pomeron models. The manuscript should include an ablation or sensitivity analysis showing how the recovered value depends on the embedding weight versus data fidelity to rule out circularity with the fitted result.
minor comments (2)
  1. [Abstract] The abstract mentions 'higher-twist scale-breaking effects through the evolution of resonance mass spectra' but does not specify how these are quantified or compared to standard higher-twist parametrizations in DIS analyses.
  2. Clarify the precise form of the string diffusion kernel and its discretization within the neural network graph, including any approximation schemes used for the five-dimensional bulk equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment in detail below and have made revisions to incorporate the suggested clarifications and analyses.

read point-by-point responses

  1. Referee: The assertion that embedding the AdS5 Dirac equation 'strictly constrains' the network to Mp ≡ 0.938 GeV requires explicit demonstration that this is a hard constraint (e.g., via architectural enforcement or infinite penalty weight) rather than a finite-weighted residual term in the loss function. Without reporting the residual norm on the Dirac operator or sensitivity tests to the embedding hyperparameter, the recovered α0 ≈ 1.0786 and x ≈ 0.19 crossover may partly reflect training dynamics or network biases instead of pure holographic QCD.

    Authors: We thank the referee for highlighting this important point regarding the nature of the constraint. The embedding is implemented via direct architectural integration of the AdS5 Dirac equation solution as a fixed layer in the network, enforcing Mp exactly at 0.938 GeV by construction. In the revised manuscript, we have added explicit reporting of the Dirac operator residual norm, which is maintained at machine precision levels (∼10^{-12}), along with sensitivity analyses to the embedding hyperparameters. These tests confirm that the reported values for α0 and the crossover point are robust and not artifacts of training dynamics. revision: yes

  2. Referee: The claim that the optimization 'naturally recovers' the Pomeron intercept α0 ≈ 1.0786 is load-bearing for the interpretation of unbiased extraction, yet α0 is a standard free parameter in holographic Pomeron models. The manuscript should include an ablation or sensitivity analysis showing how the recovered value depends on the embedding weight versus data fidelity to rule out circularity with the fitted result.

    Authors: We agree that demonstrating the independence from embedding weight is crucial to support the claim of natural recovery. We have performed the requested ablation study and included it in the revised results section. By varying the embedding loss weight relative to the data fidelity term over a wide range, we show that α0 converges to approximately 1.0786 as long as the mass constraint is satisfied, with minimal dependence on the exact weight value. This analysis rules out circularity and confirms that the intercept is determined by the SLAC data within the holographic framework. We have updated the abstract and discussion to reflect this additional evidence. revision: yes

Circularity Check

1 steps flagged

Fitted Pomeron intercept presented as 'natural recovery'; embedding claim reduces to optimization on data

specific steps
  1. fitted input called prediction [Abstract]
    "Furthermore, the optimization naturally recovers a Pomeron intercept of α0 ≈ 1.0786"

    α0 is a free parameter in holographic Pomeron models. The paper obtains its numerical value by minimizing the loss against SLAC data inside the PGNN; presenting the fitted value as a 'natural recovery' makes the reported result equivalent to the input fit rather than an independent prediction from the embedded AdS5 equations.

full rationale

The paper's central results (χ²/d.o.f. ≃ 0.91, crossover at x≈0.19, α0≈1.0786) are obtained by training the PGNN on SLAC DIS data. The abstract explicitly states that the network 'naturally recovers' the Pomeron intercept via optimization, yet this quantity is a standard tunable parameter in holographic Pomeron models. No independent first-principles derivation is shown; the value is the output of the fit. The claim of being 'strictly constrained' to Mp=0.938 GeV via direct embedding of the AdS5 Dirac equation is asserted but not demonstrated to be a hard constraint rather than a weighted loss term. This matches the 'fitted_input_called_prediction' pattern for the intercept and leaves the strict-constraint claim without supporting equations showing exact enforcement independent of training.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard holographic QCD assumptions plus data-driven fitting; no new entities are postulated.

free parameters (2)
  • Pomeron intercept α0 = 1.0786
    Recovered through optimization on SLAC data; central to the t-channel contribution.
  • neural network weights and biases
    Trained to minimize chi-squared on the structure function data while satisfying the embedded equations.
axioms (2)
  • domain assumption The five-dimensional AdS5 Dirac equation accurately describes bulk fermion dynamics for proton structure in holographic QCD.
    Directly embedded into the computational graph as the s-channel mechanism.
  • domain assumption The string diffusion kernel models t-channel holographic Pomeron exchange.
    Embedded into the network to represent the diffractive background.

pith-pipeline@v0.9.0 · 5542 in / 1653 out tokens · 69879 ms · 2026-05-13T18:41:33.893934+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 5 internal anchors

  1. [1]

    Probing Proton Structure via Physics-Guided Neural Networks in Holographic QCD

    ), including the global analysis of parton distribu- tion functions (PDFs) [54–57] and the solving of mul- tiquark bound states [58]. Beyond these, neural net- works have been successfully deployed to identify char- acteristic signals of QCD phase transitions in heavy-ion collision experiments [36, 59, 60]. In the realm of lat- tice QCD, generative models...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    E. D. Bloomet al., High-Energy Inelastic e p Scattering at 6-Degrees and 10-Degrees, Phys. Rev. Lett.23, 930 (1969)

    work page 1969

  3. [3]

    Breidenbach, J

    M. Breidenbach, J. I. Friedman, H. W. Kendall, E. D. Bloom, D. H. Coward, H. C. DeStaebler, J. Drees, L. W. Mo, and R. E. Taylor, Observed behavior of highly in- elastic electron-proton scattering, Phys. Rev. Lett.23, 935 (1969)

    work page 1969

  4. [4]

    L. W. Whitlow, S. Rock, A. Bodek, E. M. Riordan, and S. Dasu, A Precise extraction of R = sigma-L / sigma- T from a global analysis of the SLAC deep inelastic e p and e d scattering cross-sections, Phys. Lett. B250, 193 (1990)

    work page 1990

  5. [5]

    L. W. Whitlow, E. M. Riordan, S. Dasu, S. Rock, and A. Bodek, Precise measurements of the proton and deuteron structure functions from a global analysis of the SLAC deep inelastic electron scattering cross-sections, Phys. Lett. B282, 475 (1992)

    work page 1992

  6. [6]

    S. J. Brodsky and G. P. Lepage, Exclusive Processes in Quantum Chromodynamics, Adv. Ser. Direct. High En- ergy Phys.5, 93 (1989)

    work page 1989

  7. [7]

    A. v. Radyushkin, ANALYSIS OF THE HARD INCLU- 10 SIVE PROCESSES IN QUANTUM CHROMODYNAM- ICS. (IN RUSSIAN), Fiz. Elem. Chast. Atom. Yadra14, 58 (1983)

    work page 1983

  8. [8]

    R. L. Jaffe and M. Soldate, Twist Four in Electropro- duction: Canonical Operators and Coefficient Functions, Phys. Rev. D26, 49 (1982)

    work page 1982

  9. [9]

    R. K. Ellis, W. Furmanski, and R. Petronzio, Power Cor- rections to the Parton Model in QCD, Nucl. Phys. B207, 1 (1982)

    work page 1982

  10. [10]

    C. D. Roberts and A. G. Williams, Dyson-Schwinger equations and their application to hadronic physics, Prog. Part. Nucl. Phys.33, 477 (1994), arXiv:hep- ph/9403224

    work page arXiv 1994

  11. [11]

    Alkofer and L

    R. Alkofer and L. von Smekal, The Infrared behavior of QCD Green’s functions: Confinement dynamical symme- try breaking, and hadrons as relativistic bound states, Phys. Rept.353, 281 (2001), arXiv:hep-ph/0007355

    work page arXiv 2001

  12. [12]

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

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  13. [13]

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

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  14. [14]

    Anti De Sitter Space And Holography

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

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  15. [15]

    Karch, E

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

    work page arXiv 2006

  16. [16]

    Andreev and V

    O. Andreev and V. I. Zakharov, Heavy-quark poten- tials and AdS/QCD, Phys. Rev. D74, 025023 (2006), arXiv:hep-ph/0604204

    work page arXiv 2006

  17. [17]

    Polchinski and M

    J. Polchinski and M. J. Strassler, Deep inelastic scatter- ing and gauge / string duality, JHEP05, 012, arXiv:hep- th/0209211

    work page arXiv

  18. [18]

    N. R. F. Braga and A. Vega, Deep inelastic scattering of baryons in a modified soft wall model, Eur. Phys. J. C 72, 2236 (2012), arXiv:1110.2548 [hep-ph]

    work page arXiv 2012

  19. [19]

    Folco Capossoli, M

    E. Folco Capossoli, M. A. Mart´ ın Contreras, D. Li, A. Vega, and H. Boschi-Filho, Proton structure functions from an AdS/QCD model with a deformed background, Phys. Rev. D102, 086004 (2020), arXiv:2007.09283 [hep- ph]

    work page arXiv 2020

  20. [20]

    A. d. C. P. d. Nascimento, H. Boschi-Filho, and J. Noronha, Proton Structure Functions from Holographic Einstein-Dilaton Models, (2025), arXiv:2505.13649 [hep-th]

    work page arXiv 2025

  21. [21]

    C. A. Ballon Bayona, H. Boschi-Filho, and N. R. F. Braga, Deep inelastic structure functions from supergrav- ity at small x, JHEP10, 088, arXiv:0712.3530 [hep-th]

    work page arXiv

  22. [22]

    Folco Capossoli and H

    E. Folco Capossoli and H. Boschi-Filho, Deep Inelastic Scattering in the Exponentially Small Bjorken Parameter Regime from the Holographic Softwall Model, Phys. Rev. D92, 126012 (2015), arXiv:1509.01761 [hep-th]

    work page arXiv 2015

  23. [23]

    Watanabe, T

    A. Watanabe, T. Sawada, and M. Huang, Extraction of gluon distributions from structure functions at small x in holographic QCD, Phys. Lett. B805, 135470 (2020), arXiv:1910.10008 [hep-ph]

    work page arXiv 2020

  24. [24]

    A. J. Larkoski, I. Moult, and B. Nachman, Jet Substruc- ture at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning, Phys. Rept. 841, 1 (2020), arXiv:1709.04464 [hep-ph]

    work page arXiv 2020

  25. [25]

    Guest, K

    D. Guest, K. Cranmer, and D. Whiteson, Deep Learning and its Application to LHC Physics, Ann. Rev. Nucl. Part. Sci.68, 161 (2018), arXiv:1806.11484 [hep-ex]

    work page arXiv 2018

  26. [26]

    Radovic, M

    A. Radovic, M. Williams, D. Rousseau, M. Kagan, D. Bonacorsi, A. Himmel, A. Aurisano, K. Terao, and T. Wongjirad, Machine learning at the energy and inten- sity frontiers of particle physics, Nature560, 41 (2018)

    work page 2018

  27. [27]

    Albertssonet al., Machine Learning in High Energy Physics Community White Paper, J

    K. Albertssonet al., Machine Learning in High Energy Physics Community White Paper, J. Phys. Conf. Ser. 1085, 022008 (2018), arXiv:1807.02876 [physics.comp- ph]

    work page arXiv 2018

  28. [28]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, Machine learning and the physical sciences, Rev. Mod. Phys.91, 045002 (2019), arXiv:1903.10563 [physics.comp-ph]

    work page arXiv 2019

  29. [29]

    Bourilkov, Machine and Deep Learning Applications in Particle Physics, Int

    D. Bourilkov, Machine and Deep Learning Applications in Particle Physics, Int. J. Mod. Phys. A34, 1930019 (2020), arXiv:1912.08245 [physics.data-an]

    work page arXiv 2020

  30. [30]

    M. D. Schwartz, Modern Machine Learning and Particle Physics 10.1162/99608f92.beeb1183 (2021), arXiv:2103.12226 [hep-ph]

    work page doi:10.1162/99608f92.beeb1183 2021

  31. [31]

    Karagiorgi, G

    G. Karagiorgi, G. Kasieczka, S. Kravitz, B. Nachman, and D. Shih, Machine Learning in the Search for New Fundamental Physics, (2021), arXiv:2112.03769 [hep- ph]

    work page arXiv 2021

  32. [32]

    Boehnleinet al., Colloquium: Machine learning in nuclear physics, Rev

    A. Boehnleinet al., Colloquium: Machine learning in nuclear physics, Rev. Mod. Phys.94, 031003 (2022), arXiv:2112.02309 [nucl-th]

    work page arXiv 2022

  33. [33]

    Shanahan et al.,Snowmass 2021 Computational Frontier CompF03 Topical Group Report: Machine Learning,2209.07559

    P. Shanahanet al., Snowmass 2021 Computational Fron- tier CompF03 Topical Group Report: Machine Learning, (2022), arXiv:2209.07559 [physics.comp-ph]

    work page arXiv 2021

  34. [34]

    Z. Yang, Y. He, W. Chen, W.-Y. Ke, L.-G. Pang, and X.-N. Wang, Deep learning assisted jet tomography for the study of Mach cones in QGP, Eur. Phys. J. C83, 652 (2023), arXiv:2206.02393 [nucl-th]

    work page arXiv 2023

  35. [35]

    Li, H.-L

    F.-P. Li, H.-L. L¨ u, L.-G. Pang, and G.-Y. Qin, Deep-learning quasi-particle masses from QCD equa- tion of state, Phys. Lett. B844, 138088 (2023), arXiv:2211.07994 [hep-ph]

    work page arXiv 2023

  36. [36]

    He, Y.-G

    W.-B. He, Y.-G. Ma, L.-G. Pang, H.-C. Song, and K. Zhou, High-energy nuclear physics meets machine learning, Nucl. Sci. Tech.34, 88 (2023), arXiv:2303.06752 [hep-ph]

    work page arXiv 2023

  37. [37]

    K. Zhou, L. Wang, L.-G. Pang, and S. Shi, Explor- ing QCD matter in extreme conditions with Machine Learning, Prog. Part. Nucl. Phys.135, 104084 (2024), arXiv:2303.15136 [hep-ph]

    work page arXiv 2024

  38. [38]

    K. Zhou, L. Pang, S. Shi, and H. Stoecker, Deep Learn- ing for inverse problems in nuclear physics, PoSFAIR- ness2022, 064 (2023)

    work page 2023

  39. [39]

    Pang, Studying high-energy nuclear physics with machine learning, Int

    L.-G. Pang, Studying high-energy nuclear physics with machine learning, Int. J. Mod. Phys. E33, 2430009 (2024)

    work page 2024

  40. [40]

    Ma, L.-G

    Y.-G. Ma, L.-G. Pang, R. Wang, and K. Zhou, Phase Transition Study Meets Machine Learning, Chin. Phys. Lett.40, 122101 (2023), arXiv:2311.07274 [nucl-th]

    work page arXiv 2023

  41. [41]

    O.-Y. Luo, X. Chen, F.-P. Li, X.-H. Li, and K. Zhou, Neural Network Modeling of Heavy-Quark Potential from Holography, (2024), arXiv:2408.03784 [hep-ph]

    work page arXiv 2024

  42. [42]

    B. Chen, X. Chen, X. Li, Z.-R. Zhu, and K. Zhou, Exploring Transport Properties of Quark-Gluon Plasma with a Machine-Learning assisted Holographic Approach, (2024), arXiv:2404.18217 [hep-ph]

    work page arXiv 2024

  43. [43]

    Omana Kuttan, J

    M. Omana Kuttan, J. Steinheimer, K. Zhou, A. Redel- bach, and H. Stoecker, Extraction of global event features in heavy-ion collision experiments using PointNet, PoS FAIRness2022, 040 (2023). 11

    work page 2023

  44. [44]

    S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, From lattice QCD to in-medium heavy-quark inter- actions via deep learning, PoSLATTICE2021, 537 (2022)

    work page 2022

  45. [45]

    S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, From lattice QCD to in-medium heavy-quark interac- tions via deep learning, EPJ Web Conf.259, 04003 (2022)

    work page 2022

  46. [46]

    S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, Heavy quark potential in the quark-gluon plasma: Deep neural network meets lattice quantum chromodynamics, Phys. Rev. D105, 014017 (2022), arXiv:2105.07862 [hep- ph]

    work page arXiv 2022

  47. [47]

    Mansouri, K

    M. Mansouri, K. Bitaghsir Fadafan, and X. Chen, Holo- graphic complex potential of a quarkonium from deep learning, (2024), arXiv:2406.06285 [hep-ph]

    work page arXiv 2024

  48. [48]

    Chen and M

    X. Chen and M. Huang, Flavor dependent Critical end- point from holographic QCD through machine learning, (2024), arXiv:2405.06179 [hep-ph]

    work page arXiv 2024

  49. [49]

    Chen and M

    X. Chen and M. Huang, Machine learning holographic black hole from lattice QCD equation of state, Phys. Rev. D109, L051902 (2024), arXiv:2401.06417 [hep-ph]

    work page arXiv 2024

  50. [50]

    X.-Y. Wang, C. Dong, and X. Liu, Analysis of Strong Coupling Constant with Machine Learning and Its Application, Chin. Phys. Lett.41, 031201 (2023), arXiv:2304.07682 [hep-ph]

    work page arXiv 2023

  51. [51]

    M. P. Bento, H. B. Cˆ amara, and J. F. Seabra, Un- raveling particle dark matter with Physics-Informed Neural Networks, Phys. Lett. B868, 139690 (2025), arXiv:2502.17597 [hep-ph]

    work page arXiv 2025

  52. [52]

    Gao, Z.-B

    M. Gao, Z.-B. Kang, J. Penttala, and D. Y. Shao, Determination of the initial condition for the Balitsky-Kovchegov equation with transformers, (2025), arXiv:2510.26779 [hep-ph]

    work page arXiv 2025

  53. [53]

    T. I. Baihaqi, C. Setyadi, Z. Akbar, P. T. P. Hutauruk, and A. Salim Adam, Determination of the HERA coher- ent diffractiveJ/ψproduction cross section via artificial neural network, (2025), arXiv:2512.21704 [hep-ph]

    work page arXiv 2025

  54. [54]

    Aarts, K

    G. Aarts, K. Fukushima, T. Hatsuda, A. Ipp, S. Shi, L. Wang, and K. Zhou, Physics-driven learning for in- verse problems in quantum chromodynamics, Nature Rev. Phys.7, 154 (2025), arXiv:2501.05580 [hep-lat]

    work page arXiv 2025

  55. [55]

    Del Debbio, S

    L. Del Debbio, S. Forte, J. I. Latorre, A. Piccione, and J. Rojo (NNPDF), Unbiased determination of the pro- ton structure function F(2)**p with faithful uncertainty estimation, JHEP03, 080, arXiv:hep-ph/0501067

    work page arXiv

  56. [56]

    R. D. Ballet al., Parton distributions with LHC data, Nucl. Phys. B867, 244 (2013), arXiv:1207.1303 [hep-ph]

    work page Pith review arXiv 2013

  57. [57]

    R. D. Ballet al.(NNPDF), Parton distributions for the LHC Run II, JHEP04, 040, arXiv:1410.8849 [hep-ph]

    work page Pith review arXiv

  58. [58]

    R. D. Ballet al.(NNPDF), The path to proton struc- ture at 1% accuracy, Eur. Phys. J. C82, 428 (2022), arXiv:2109.02653 [hep-ph]

    work page arXiv 2022

  59. [59]

    W.-L. Wu, L. Meng, and S.-L. Zhu, DeepQuark: A Deep- Neural-Network Approach to Multiquark Bound States, Phys. Rev. Lett.136, 071901 (2026), arXiv:2506.20555 [hep-ph]

    work page arXiv 2026

  60. [60]

    Wang and Q

    Y. Wang and Q. Li, Machine learning transforms the inference of the nuclear equation of state, Front. Phys. (Beijing)18, 64402 (2023), arXiv:2305.16686 [nucl-th]

    work page arXiv 2023

  61. [61]

    R. Wang, C. Qiu, C.-S. Hu, Z. Li, and Y. Wu, Identifying weak critical fluctuations of intermittency in heavy-ion collisions with topological machine learning, Phys. Lett. B864, 139405 (2025), arXiv:2412.06151 [nucl-th]

    work page arXiv 2025

  62. [62]

    Kanwar, M

    G. Kanwar, M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, S. Racani` ere, D. J. Rezende, and P. E. Shanahan, Equivariant flow-based sampling for lattice gauge theory, Phys. Rev. Lett.125, 121601 (2020), arXiv:2003.06413 [hep-lat]

    work page arXiv 2020

  63. [63]

    de Oliveira, M

    L. de Oliveira, M. Kagan, L. Mackey, B. Nachman, and A. Schwartzman, Jet-images — deep learning edition, JHEP07, 069, arXiv:1511.05190 [hep-ph]

    work page arXiv

  64. [64]

    Qu and L

    H. Qu and L. Gouskos, ParticleNet: Jet Tagging via Particle Clouds, Phys. Rev. D101, 056019 (2020), arXiv:1902.08570 [hep-ph]

    work page arXiv 2020

  65. [65]

    W. Kou, X. Lin, B. Guo, and X. Chen, Physics-informed neural network approach to quark-antiquark color flux tube, Phys. Rev. D112, 056027 (2025), arXiv:2506.03513 [hep-ph]

    work page arXiv 2025

  66. [66]

    Extraction of the color dipole amplitude with physics-informed neural networks

    W. Kou and X. Chen, Universality of Gluon Satura- tion from Physics-Informed Neural Networks, (2026), arXiv:2601.16391 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  67. [67]

    Gutsche, V

    T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Dilaton in a soft-wall holographic approach to mesons and baryons, Phys. Rev. D85, 076003 (2012), arXiv:1108.0346 [hep-ph]

    work page arXiv 2012

  68. [68]

    Gao and B.-W

    J.-H. Gao and B.-W. Xiao, Polarized Deep Inelastic and Elastic Scattering From Gauge/String Duality, Phys. Rev. D80, 015025 (2009), arXiv:0904.2870 [hep-ph]

    work page arXiv 2009

  69. [69]

    Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics- informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys.378, 686 (2019), arXiv:1711.10561 [cs.AI]

    work page Pith review arXiv 2019

  70. [70]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learn- ing, Nature Reviews Physics3, 422 (2021)

    work page 2021

  71. [71]

    Abidin and C

    Z. Abidin and C. E. Carlson, Nucleon electromagnetic and gravitational form factors from holography, Phys. Rev. D79, 115003 (2009), arXiv:0903.4818 [hep-ph]

    work page arXiv 2009

  72. [72]

    Donnachie and P

    A. Donnachie and P. V. Landshoff, Total cross-sections, Phys. Lett. B296, 227 (1992), arXiv:hep-ph/9209205

    work page arXiv 1992

  73. [73]

    Combination of Measurements of Inclusive Deep Inelastic $e^{\pm}p$ Scattering Cross Sections and QCD Analysis of HERA Data

    H. Abramowiczet al.(H1, ZEUS), Combination of mea- surements of inclusive deep inelastice ±pscattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C75, 580 (2015), arXiv:1506.06042 [hep-ex]

    work page Pith review arXiv 2015

  74. [74]

    E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo (NNPDF), A first unbiased global determination of po- larized PDFs and their uncertainties, Nucl. Phys. B887, 276 (2014), arXiv:1406.5539 [hep-ph]

    work page Pith review arXiv 2014

  75. [75]

    J. J. Ethier, N. Sato, and W. Melnitchouk, First simulta- neous extraction of spin-dependent parton distributions and fragmentation functions from a global QCD analysis, Phys. Rev. Lett.119, 132001 (2017), arXiv:1705.05889 [hep-ph]

    work page Pith review arXiv 2017

  76. [76]

    A. Vega, I. Schmidt, T. Gutsche, and V. E. Lyubovitskij, Generalized parton distributions in AdS/QCD, Phys. Rev. D83, 036001 (2011), arXiv:1010.2815 [hep-ph]

    work page arXiv 2011

  77. [77]

    ˇCui´ c, K

    M. ˇCui´ c, K. Kumeriˇ cki, and A. Sch¨ afer, Separation of Quark Flavors Using Deeply Virtual Compton Scat- tering Data, Phys. Rev. Lett.125, 232005 (2020), arXiv:2007.00029 [hep-ph]

    work page arXiv 2020