Artificial neural network in cosmic landscape


In this paper we propose that artificial neural network, the basis of machine learning, is useful to generate the inflationary landscape from a cosmological point of view. Traditional numerical simulations of a global cosmic landscape typically need an exponential complexity when the number of fields is large. However, a basic application of artificial neural network could solve the problem based on the universal approximation theorem of the multilayer perceptron. A toy model in inflation with multiple light fields is investigated numerically as an example of such an application.

A preprint version of the article is available at ArXiv.


  1. [1]

    A.H. Guth, The inflationary universe: a possible solution to the horizon and flatness problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].

  2. [2]

    A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108 (1982) 389.

  3. [3]

    A. Albrecht and P.J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48 (1982) 1220 [INSPIRE].

    Article  ADS  Google Scholar 

  4. [4]

    J. Martin, C. Ringeval and V. Vennin, Encyclopædia inflationaris, Phys. Dark Univ. 5-6 (2014) 75 [arXiv:1303.3787] [INSPIRE].

  5. [5]

    P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  6. [6]

    S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].

  7. [7]

    L. Susskind, The Anthropic landscape of string theory, in Universe or multiverse?, B. Carr ed., Cambridge University Press, Cambridge U.K. (2009), hep-th/0302219 [INSPIRE].

  8. [8]

    M.R. Douglas, Statistics of string vacua, hep-ph/0401004 [INSPIRE].

  9. [9]

    M.R. Douglas, Basic results in vacuum statistics, Compt. Rend. Phys. 5 (2004) 965 [hep-th/0409207] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  10. [10]

    C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].

  11. [11]

    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  12. [12]

    M.R. Douglas, Understanding the landscape, hep-th/0602266 [INSPIRE].

  13. [13]

    A. Berera, Thermal properties of an inflationary universe, Phys. Rev. D 54 (1996) 2519 [hep-th/9601134] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    Q.-G. Huang and S.H.H. Tye, The cosmological constant problem and inflation in the string landscape, Int. J. Mod. Phys. A 24 (2009) 1925 [arXiv:0803.0663] [INSPIRE].

  15. [15]

    S.H.H. Tye and J. Xu, A meandering inflaton, Phys. Lett. B 683 (2010) 326 [arXiv:0910.0849] [INSPIRE].

    Article  ADS  Google Scholar 

  16. [16]

    D. Battefeld, T. Battefeld, C. Byrnes and D. Langlois, Beauty is distractive: particle production during multifield inflation, JCAP 08 (2011) 025 [arXiv:1106.1891] [INSPIRE].

    Article  ADS  Google Scholar 

  17. [17]

    M. Dias, J. Frazer and A.R. Liddle, Multifield consequences for D-brane inflation, JCAP 06 (2012) 020 [Erratum ibid. 03 (2013) E01] [arXiv:1203.3792] [INSPIRE].

  18. [18]

    D. Green, Disorder in the early universe, JCAP 03 (2015) 020 [arXiv:1409.6698] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  19. [19]

    J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay, The statistics of peaks of gaussian random fields, Astrophys. J. 304 (1986) 15 [INSPIRE].

    Article  ADS  Google Scholar 

  20. [20]

    R. Easther, A.H. Guth and A. Masoumi, Counting vacua in random landscapes, arXiv:1612.05224 [INSPIRE].

  21. [21]

    J. Liu, Y. Wang and S. Zhou, Nonuniqueness of classical inflationary trajectories on a high-dimensional landscape, Phys. Rev. D 91 (2015) 103525 [arXiv:1501.06785] [INSPIRE].

    ADS  Google Scholar 

  22. [22]

    A. Aazami and R. Easther, Cosmology from random multifield potentials, JCAP 03 (2006) 013 [hep-th/0512050] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  23. [23]

    M.C.D. Marsh, L. McAllister, E. Pajer and T. Wrase, Charting an inflationary landscape with random matrix theory, JCAP 11 (2013) 040 [arXiv:1307.3559] [INSPIRE].

    Article  ADS  Google Scholar 

  24. [24]

    T. Battefeld and C. Modi, Local random potentials of high differentiability to model the Landscape, JCAP 03 (2015) 010 [arXiv:1409.5135] [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

  25. [25]

    T.C. Bachlechner, On gaussian random supergravity, JHEP 04 (2014) 054 [arXiv:1401.6187] [INSPIRE].

    MathSciNet  Article  MATH  ADS  Google Scholar 

  26. [26]

    M. Tegmark, What does inflation really predict?, JCAP 04 (2005) 001 [astro-ph/0410281] [INSPIRE].

  27. [27]

    J. Frazer and A.R. Liddle, Multi-field inflation with random potentials: field dimension, feature scale and non-Gaussianity, JCAP 02 (2012) 039 [arXiv:1111.6646] [INSPIRE].

    Article  ADS  Google Scholar 

  28. [28]

    F. Duplessis, Y. Wang and R. Brandenberger, Multi-stream inflation in a landscape, JCAP 04 (2012) 012 [arXiv:1201.0029] [INSPIRE].

    Article  ADS  Google Scholar 

  29. [29]

    S. Russell and P. Norvig, Artificial Intelligence: a modern approach, Pearson, U.S.A. (2009).

  30. [30]

    S. Haykin, Neural networks, a comprehensive foundation, Prentice Hall, U.S.A. (1999).

  31. [31]

    M. Hassoun, Fundamentals of artificial neural networks, Bradford Books, U.S.A. (2003).

  32. [32]

    W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biol. 5 (1943) 115.

    MathSciNet  MATH  Google Scholar 

  33. [33]

    G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control. Sign. Syst. 2 (1989) 303.

    MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Network 4 (1991) 251.

    Article  Google Scholar 

  35. [35]

    D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].

  36. [36]

    F. Denef and M.R. Douglas, Computational complexity of the landscape. I, Annals Phys. 322 (2007) 1096 [hep-th/0602072] [INSPIRE].

  37. [37]

    F. Denef, M.R. Douglas, B. Greene and C. Zukowski, Computational complexity of the landscape. II. Cosmological considerations, arXiv:1706.06430 [INSPIRE].

  38. [38]

    N. Bao, R. Bousso, S. Jordan and B. Lackey, Fast optimization algorithms and the cosmological constant, Phys. Rev. D 96 (2017) 103512 [arXiv:1706.08503] [INSPIRE].

    ADS  Google Scholar 

  39. [39]

    J. Carrasquilla and R. Melko, Machine learning phases of matter, Nature Phys. 13 (2017) 431 [arXiv:1605.01735].

    Article  ADS  Google Scholar 

  40. [40]

    ATLAS collaboration, A neural network clustering algorithm for the ATLAS silicon pixel detector, 2014 JINST 9 P09009 [arXiv:1406.7690] [INSPIRE].

  41. [41]

    L. Sagun et al., Explorations on high dimensional landscapes, arXiv:1412.6615.

  42. [42]

    P. Chaudhari and S. Stefano, On the energy landscape of deep networks, arXiv:1511.06485.

  43. [43]

    A.J. Ballard et al., Perspective: energy landscapes for machine learning, Phys. Chem. Chem. Phys. 19 (2017) 2585 [arXiv:1703.07915] [INSPIRE].

    Article  Google Scholar 

  44. [44]

    T. Cohen, M. Freytsis and B. Ostdiek, (Machine) learning to do more with less, arXiv:1706.09451 [INSPIRE].

  45. [45]

    Y. Huang and J. Moore, Neural network representation of tensor network and chiral states, arXiv:1701.06246.

  46. [46]

    C.P. Novaes, A. Bernui, I.S. Ferreira and C.A. Wuensche, A neural-network based estimator to search for primordial non-Gaussianity in Planck CMB maps, JCAP 09 (2015) 064 [arXiv:1409.3876] [INSPIRE].

    Article  ADS  Google Scholar 

  47. [47]

    Y.-H. He, Deep-learning the landscape, arXiv:1706.02714 [INSPIRE].

  48. [48]

    D. Krefl and R.-K. Seong, Machine learning of Calabi-Yau volumes, Phys. Rev. D 96 (2017) 066014 [arXiv:1706.03346] [INSPIRE].

  49. [49]

    F. Ruehle, Evolving neural networks with genetic algorithms to study the string landscape, JHEP 08 (2017) 038 [arXiv:1706.07024] [INSPIRE].

    MathSciNet  Article  MATH  ADS  Google Scholar 

  50. [50]

    J. Carifio, J. Halverson, D. Krioukov and B.D. Nelson, Machine learning in the string landscape, JHEP 09 (2017) 157 [arXiv:1707.00655] [INSPIRE].

    MathSciNet  Article  MATH  ADS  Google Scholar 

  51. [51]

    G. Flake, Nonmonotonic activation functions in multilayer perceptrons, Ph.D. Thesis, University of Maryland, College Park, U.S.A. (1993).

  52. [52]

    J. Martin and R.H. Brandenberger, The transplanckian problem of inflationary cosmology, Phys. Rev. D 63 (2001) 123501 [hep-th/0005209] [INSPIRE].

    ADS  Google Scholar 

  53. [53]

    D.H. Lyth, What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy?, Phys. Rev. Lett. 78 (1997) 1861 [hep-ph/9606387] [INSPIRE].

  54. [54]

    X. Chen and Y. Wang, Quasi-single field inflation and non-gaussianities, JCAP 04 (2010) 027 [arXiv:0911.3380] [INSPIRE].

    Article  ADS  Google Scholar 

  55. [55]

    J. Diestel and A .Spalsbury, The joys of Haar measure, American Mathematical Society, Providence U.S.A. (2014).

  56. [56]

    A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Ann. Math. 34 (1933) 147.

    MathSciNet  Article  MATH  Google Scholar 

  57. [57]

    A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].

    MathSciNet  Article  ADS  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information



Corresponding author

Correspondence to Junyu Liu.

Additional information

ArXiv ePrint: 1707.02800

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, J. Artificial neural network in cosmic landscape. J. High Energ. Phys. 2017, 149 (2017).

Download citation


  • Cosmology of Theories beyond the SM
  • Models of Quantum Gravity
  • Random Systems