Probing proton structure via physics-guided neural networks in holographic QCD

[1]
E. D. Bloom, D. H. Coward, H. DeStaebler et al., Phys. Rev. Lett. 23, 930 (1969)
[2]
M. Breidenbach, J. I. Friedman, H. W. Kendall et al., Phys. Rev. Lett. 23, 935 (1969)
[3]
L. W. Whitlow, S. Rock, A. Bodek et al., Phys. Lett. B 250, 193 (1990)
[4]
L. W. Whitlow, E. M. Riordan, S. Dasu et al., Phys. Lett. B 282, 475 (1992)
[5]
S. J. Brodsky and G. P. Lepage, Adv. Ser. Direct. High Energy Phys. 5, 93 (1989)
[6]
A. v. Radyushkin, Fiz. Elem. Chast. Atom. Yadra 14, 58 (1983)
[7]
R. L. Jaffe and M. Soldate, Phys. Rev. D 26, 49 (1982)
[8]
R. K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. B 207, 1 (1982)
[9]
C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994), arXiv: hep-ph/9403224
[10]
R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001), arXiv: hep-ph/0007355
[11]
J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv: hep-th/9711200
[12]
S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428, 105 (1998), arXiv: hep-th/9802109
[13]
E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv: hep-th/9802150
[14]
A. Karch, E. Katz, D. T. Son et al., Phys. Rev. D 74, 015005 (2006), arXiv: hep-ph/0602229
[15]
O. Andreev and V. I. Zakharov, Phys. Rev. D 74, 025023 (2006), arXiv: hep-ph/0604204
[16]
T. Gutsche, V. E. Lyubovitskij, I. Schmidt et al., Phys. Rev. D 85, 076003 (2012), arXiv: 1108.0346[hep-ph]
[17]
J. Polchinski and M. J. Strassler, JHEP 05, 012 (2003), arXiv: hep-th/0209211
[18]
N. R. F. Braga and A. Vega, Eur. Phys. J. C 72, 2236 (2012), arXiv: 1110.2548[hep-ph]
[19]
E. Folco Capossoli, M. A. Martín Contreras, D. Li et al., Phys. Rev. D 102, 086004 (2020), arXiv: 2007.09283[hep-ph]
[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]
[21]
J. Deng and D. Hou, Phys. Rev. D 112, 036011 (2025), arXiv: 2502.00771[nucl-th]
[22]
J. Deng and D. Hou, Exploring Nucleon Structure and the Proton Mass Problem through Holographic QCD, (2026), arXiv: 2603.04794[hep-ph]
[23]
C. A. Ballon Bayona, H. Boschi-Filho, and N. R. F. Braga, JHEP 10, 088 (2008), arXiv: 0712.3530[hep-th]
[24]
E. Folco Capossoli and H. Boschi-Filho, Phys. Rev. D 92, 126012 (2015), arXiv: 1509.01761[hep-th]
[25]
A. Watanabe, T. Sawada, and M. Huang, Phys. Lett. B 805, 135470 (2020), arXiv: 1910.10008[hep-ph]
[26]
A. J. Larkoski, I. Moult, and B. Nachman, Phys. Rept. 841, 1 (2020), arXiv: 1709.04464[hep-ph]
[27]
D. Guest, K. Cranmer, and D. Whiteson, Ann. Rev. Nucl. Part. Sci. 68, 161 (2018), arXiv: 1806.11484[hep-ex]
[28]
A. Radovic, M. Williams, D. Rousseau et al., Nature 560, 41 (2018)
[29]
K. Albertsson et al., J. Phys. Conf. Ser. 1085, 022008 (2018), arXiv: 1807.02876[physics.comp-ph]
[30]
G. Carleo, I. Cirac, K. Cranmer et al., Rev. Mod. Phys. 91, 045002 (2019), arXiv: 1903.10563[physics.comp-ph]
[31]
A. Boehnlein et al., Rev. Mod. Phys. 94, 031003 (2022), arXiv: 2112.02309[nucl-th]
[32]
G. Aarts, K. Fukushima, T. Hatsuda et al., Nature Rev. Phys. 7, 154 (2025), arXiv: 2501.05580[hep-lat]
[33]
L. Del Debbio et al. (NNPDF Collaboration), JHEP 03, 080 (2005), arXiv: hep-ph/0501067
[34]
R. D. Ball et al. (NNPDF Collaboration), Nucl. Phys. B 867, 244 (2013), arXiv: 1207.1303[hep-ph]
[35]
R. D. Ball et al. (NNPDF Collaboration), JHEP 04, 040 (2015), arXiv: 1410.8849[hep-ph]
[36]
R. D. Ball et al. (NNPDF Collaboration), Eur. Phys. J. C 77, 663 (2017), arXiv: 1706.00428[hep-ph]
[37]
S. Carrazza and J. Cruz-Martinez, Eur. Phys. J. C 79, 676 (2019), arXiv: 1907.05075[hep-ph]
[38]
R. D. Ball et al. (NNPDF Collaboration), Eur. Phys. J. C 82, 428 (2022), arXiv: 2109.02653[hep-ph]
[39]
G. Kanwar, M. S. Albergo, D. Boyda et al., Phys. Rev. Lett. 125, 121601 (2020), arXiv: 2003.06413[hep-lat]
[40]
M. Raissi, P. Perdikaris, and G. E. Karniadakis, J. Comput. Phys. 378, 686 (2019), arXiv: 1711.10561[cs.AI]
[41]
G. E. Karniadakis, I. G. Kevrekidis, L. Lu et al., Nature Rev. Phys. 3, 422 (2021)
[42]
W. Kou, X. Lin, B. Guo et al., Phys. Rev. D 112, 056027 (2025), arXiv: 2506.03513[hep-ph]
[43]
W. Kou and X. Chen, Universality of Gluon Saturation from Physics-Informed Neural Networks, (2026), arXiv: 2601.16391[hep-ph]
[44]
S. W. Dai, F. P. Li, L. G. Pang et al., Physics-Informed Global Extraction of the Universal Small-x Dipole Amplitude, (2026), arXiv: 2603.08008[hep-ph]
[45]
M. Gao, Z. B. Kang, J. Penttala et al., Determination of the initial condition for the Balitsky-Kovchegov equation with transformers, (2025), arXiv: 2510.26779[hep-ph]
[46]
T. I. Baihaqi, C. Setyadi, Z. Akbar et al., Determination of the HERA coherent diffractive \begin{document}$J/\psi$\end{document} production cross section via artificial neural network, (2025), arXiv: 2512.21704[hep-ph]
[47]
A. Donnachie and P. V. Landshoff, Phys. Lett. B 296, 227 (1992), arXiv: hep-ph/9209205
[48]
T. Gutsche, V. E. Lyubovitskij, and I. Schmidt, Phys. Rev. D 97, 054011 (2018), arXiv: 1712.08410[hep-ph]
[49]
R. C. Brower, J. Polchinski, M. J. Strassler et al., JHEP 12, 005 (2007), arXiv: hep-th/0603115
[50]
R. C. Brower, M. Djuric, I. Sarcevic et al., JHEP 11, 051 (2010), arXiv: 1007.2259[hep-ph]
[51]
C. A. Ballon Bayona, H. Boschi-Filho, and N. R. F. Braga, JHEP 03, 064 (2008), arXiv: 0711.0221[hep-th]
[52]
Z. Abidin and C. E. Carlson, Phys. Rev. D 79, 115003 (2009), arXiv: 0903.4818[hep-ph]
[53]
J. H. Gao and B. W. Xiao, Phys. Rev. D 80, 015025 (2009), arXiv: 0904.2870[hep-ph]
[54]
S. J. Brodsky, G. F. de Teramond, H. G. Dosch et al., Phys. Rept. 584, 1 (2015), arXiv: 1407.8131[hep-ph]
[55]
S. Mamedov and S. Taghiyeva, J. Korean Phys. Soc. 85, 541 (2024), arXiv: 2208.07115[hep-ph]
[56]
Y. Hatta, E. Iancu, and A. H. Mueller, JHEP 01, 026 (2008), arXiv: 0710.2148[hep-th]
[57]
D. Jorrin and M. Schvellinger, Phys. Rev. D 106, 066024 (2022), arXiv: 2207.02984[hep-ph]
[58]
I. Borsa, D. Jorrin, R. Sassot et al., Phys. Rev. D 108, 056024 (2023), arXiv: 2308.01975[hep-ph]
[59]
A. Karpatne, G. Atluri, J. H. Faghmous et al., IEEE Trans. Knowl. Data Eng. 29, 2318 (2017)
[60]
J. Willard, X. Jia, S. Xu et al., ACM Computing Surveys 55, 1 (2022)
[61]
A. Raue, C. Kreutz, T. Maiwald et al., Bioinformatics 25, 1923 (2009)
[62]
A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston & Sons, Washington, DC, 1977).
[63]
D. V. Raman, J. Anderson, and A. Papachristodoulou, Phys. Rev. E 95, 032314 (2017), arXiv: 1607.07705[physics.data-an]
[64]
M. F. Kasim, D. Watson-Parris, L. Deaconu, S. Oliver, P. Hatfield, D. H. Froula, G. Gregori, M. Jarvis, S. Khatiwala, J. Korenaga, et al., Machine Learning: Science and Technology 3, 015013 (2022)
[65]
D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, in International Conference on Learning Representations (2014), arXiv: 1412.6980[cs.LG]
[66]
M. C. Kennedy and A. O'Hagan, JRSS-B 63, 425 (2001)
[67]
J. Ye, J. Am. Stat. Assoc. 93, 120 (1998)
[68]
P. J. Huber, Robust estimation of a location parameter, in Breakthroughs in statistics: Methodology and distribution (Springer, 1992) p. 492
[69]
R. Pascanu, T. Mikolov, and Y. Bengio, arXiv: 1211.5063[cs.LG]
[70]
A. de Myttenaere, B. Golden, B. Le Grand et al., Neurocomputing 192, 38 (2016)
[71]
S. Elfwing, E. Uchibe, and K. Doya, arXiv: 1702.03118[cs.LG]
[72]
H. Abramowicz et al. (H1, ZEUS), Eur. Phys. J. C 75, 580 (2015), arXiv: 1506.06042[hep-ex]
[73]
J. Dudek et al., Eur. Phys. J. A 48, 187 (2012), arXiv: 1208.1244[hep-ex]
[74]
R. Abdul Khalek et al., Nucl. Phys. A 1026, 122447 (2022), arXiv: 2103.05419[physics.ins-det]
[75]
D. P. Anderle et al., Front. Phys. (Beijing) 16, 64701 (2021), arXiv: 2102.09222[nucl-ex]
[76]
P. Agostini et al. (LHeC, FCC-he Study Group), J. Phys. G 48, 110501 (2021), arXiv: 2007.14491[hep-ex]