Abstract
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.
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References
A.H. Guth, The inflationary universe: a possible solution to the horizon and flatness problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
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.
A. Albrecht and P.J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48 (1982) 1220 [INSPIRE].
J. Martin, C. Ringeval and V. Vennin, Encyclopædia inflationaris, Phys. Dark Univ. 5-6 (2014) 75 [arXiv:1303.3787] [INSPIRE].
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
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].
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].
M.R. Douglas, Statistics of string vacua, hep-ph/0401004 [INSPIRE].
M.R. Douglas, Basic results in vacuum statistics, Compt. Rend. Phys. 5 (2004) 965 [hep-th/0409207] [INSPIRE].
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
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].
M.R. Douglas, Understanding the landscape, hep-th/0602266 [INSPIRE].
A. Berera, Thermal properties of an inflationary universe, Phys. Rev. D 54 (1996) 2519 [hep-th/9601134] [INSPIRE].
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].
S.H.H. Tye and J. Xu, A meandering inflaton, Phys. Lett. B 683 (2010) 326 [arXiv:0910.0849] [INSPIRE].
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].
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].
D. Green, Disorder in the early universe, JCAP 03 (2015) 020 [arXiv:1409.6698] [INSPIRE].
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].
R. Easther, A.H. Guth and A. Masoumi, Counting vacua in random landscapes, arXiv:1612.05224 [INSPIRE].
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].
A. Aazami and R. Easther, Cosmology from random multifield potentials, JCAP 03 (2006) 013 [hep-th/0512050] [INSPIRE].
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].
T. Battefeld and C. Modi, Local random potentials of high differentiability to model the Landscape, JCAP 03 (2015) 010 [arXiv:1409.5135] [INSPIRE].
T.C. Bachlechner, On gaussian random supergravity, JHEP 04 (2014) 054 [arXiv:1401.6187] [INSPIRE].
M. Tegmark, What does inflation really predict?, JCAP 04 (2005) 001 [astro-ph/0410281] [INSPIRE].
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].
F. Duplessis, Y. Wang and R. Brandenberger, Multi-stream inflation in a landscape, JCAP 04 (2012) 012 [arXiv:1201.0029] [INSPIRE].
S. Russell and P. Norvig, Artificial Intelligence: a modern approach, Pearson, U.S.A. (2009).
S. Haykin, Neural networks, a comprehensive foundation, Prentice Hall, U.S.A. (1999).
M. Hassoun, Fundamentals of artificial neural networks, Bradford Books, U.S.A. (2003).
W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biol. 5 (1943) 115.
G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control. Sign. Syst. 2 (1989) 303.
K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Network 4 (1991) 251.
D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
F. Denef and M.R. Douglas, Computational complexity of the landscape. I, Annals Phys. 322 (2007) 1096 [hep-th/0602072] [INSPIRE].
F. Denef, M.R. Douglas, B. Greene and C. Zukowski, Computational complexity of the landscape. II. Cosmological considerations, arXiv:1706.06430 [INSPIRE].
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].
J. Carrasquilla and R. Melko, Machine learning phases of matter, Nature Phys. 13 (2017) 431 [arXiv:1605.01735].
ATLAS collaboration, A neural network clustering algorithm for the ATLAS silicon pixel detector, 2014 JINST 9 P09009 [arXiv:1406.7690] [INSPIRE].
L. Sagun et al., Explorations on high dimensional landscapes, arXiv:1412.6615.
P. Chaudhari and S. Stefano, On the energy landscape of deep networks, arXiv:1511.06485.
A.J. Ballard et al., Perspective: energy landscapes for machine learning, Phys. Chem. Chem. Phys. 19 (2017) 2585 [arXiv:1703.07915] [INSPIRE].
T. Cohen, M. Freytsis and B. Ostdiek, (Machine) learning to do more with less, arXiv:1706.09451 [INSPIRE].
Y. Huang and J. Moore, Neural network representation of tensor network and chiral states, arXiv:1701.06246.
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].
Y.-H. He, Deep-learning the landscape, arXiv:1706.02714 [INSPIRE].
D. Krefl and R.-K. Seong, Machine learning of Calabi-Yau volumes, Phys. Rev. D 96 (2017) 066014 [arXiv:1706.03346] [INSPIRE].
F. Ruehle, Evolving neural networks with genetic algorithms to study the string landscape, JHEP 08 (2017) 038 [arXiv:1706.07024] [INSPIRE].
J. Carifio, J. Halverson, D. Krioukov and B.D. Nelson, Machine learning in the string landscape, JHEP 09 (2017) 157 [arXiv:1707.00655] [INSPIRE].
G. Flake, Nonmonotonic activation functions in multilayer perceptrons, Ph.D. Thesis, University of Maryland, College Park, U.S.A. (1993).
J. Martin and R.H. Brandenberger, The transplanckian problem of inflationary cosmology, Phys. Rev. D 63 (2001) 123501 [hep-th/0005209] [INSPIRE].
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].
X. Chen and Y. Wang, Quasi-single field inflation and non-gaussianities, JCAP 04 (2010) 027 [arXiv:0911.3380] [INSPIRE].
J. Diestel and A .Spalsbury, The joys of Haar measure, American Mathematical Society, Providence U.S.A. (2014).
A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Ann. Math. 34 (1933) 147.
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
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Liu, J. Artificial neural network in cosmic landscape. J. High Energ. Phys. 2017, 149 (2017). https://doi.org/10.1007/JHEP12(2017)149
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DOI: https://doi.org/10.1007/JHEP12(2017)149