Abstract
The landscape of low-energy effective field theories stemming from string theory is too vast for a systematic exploration. However, the meadows of the string landscape may be fertile ground for the application of machine learning techniques. Employing neural network learning may allow for inferring novel, undiscovered properties that consistent theories in the landscape should possess, or checking conjectural statements about alleged characteristics thereof. The aim of this work is to describe to what extent the string landscape can be explored with neural network-based learning. Our analysis is motivated by recent studies that show that the string landscape is characterized by finiteness properties, emerging from its underlying tame, o-minimal structures. Indeed, employing these results, we illustrate that any low-energy effective theory of string theory is endowed with certain statistical learnability properties. Consequently, several local learning problems therein formulated, including interpolations and multi-class classification problems, can be concretely addressed with machine learning, delivering results with sufficiently high accuracy.
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References
S. Ashok and M.R. Douglas, Counting flux vacua, JHEP 01 (2004) 060 [hep-th/0307049] [INSPIRE].
F. Denef and M.R. Douglas, Distributions of flux vacua, JHEP 05 (2004) 072 [hep-th/0404116] [INSPIRE].
F. Denef, M.R. Douglas and B. Florea, Building a better racetrack, JHEP 06 (2004) 034 [hep-th/0404257] [INSPIRE].
W. Taylor and Y.-N. Wang, The F-theory geometry with most flux vacua, JHEP 12 (2015) 164 [arXiv:1511.03209] [INSPIRE].
D. Krefl and R.-K. Seong, Machine Learning of Calabi-Yau Volumes, Phys. Rev. D 96 (2017) 066014 [arXiv:1706.03346] [INSPIRE].
R. Altman, J. Carifio, J. Halverson and B.D. Nelson, Estimating Calabi-Yau Hypersurface and Triangulation Counts with Equation Learners, JHEP 03 (2019) 186 [arXiv:1811.06490] [INSPIRE].
C.R. Brodie, A. Constantin, R. Deen and A. Lukas, Machine Learning Line Bundle Cohomology, Fortsch. Phys. 68 (2020) 1900087 [arXiv:1906.08730] [INSPIRE].
Y.-H. He and A. Lukas, Machine Learning Calabi-Yau Four-folds, Phys. Lett. B 815 (2021) 136139 [arXiv:2009.02544] [INSPIRE].
L.B. Anderson et al., Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning, JHEP 05 (2021) 013 [arXiv:2012.04656] [INSPIRE].
M.R. Douglas, S. Lakshminarasimhan and Y. Qi, Numerical Calabi-Yau metrics from holomorphic networks, arXiv:2012.04797 [INSPIRE].
M.R. Douglas, Holomorphic feedforward networks, Pure Appl. Math. Quart. 18 (2022) 251 [arXiv:2105.03991] [INSPIRE].
M.R. Douglas, From Algebraic Geometry to Machine Learning, arXiv:2107.14387 [INSPIRE].
A. Constantin, T.R. Harvey and A. Lukas, Heterotic String Model Building with Monad Bundles and Reinforcement Learning, Fortsch. Phys. 70 (2022) 2100186 [arXiv:2108.07316] [INSPIRE].
M. Larfors, A. Lukas, F. Ruehle and R. Schneider, Learning Size and Shape of Calabi-Yau Spaces, arXiv:2111.01436 [INSPIRE].
S.A. Abel et al., Decoding Nature with Nature’s Tools: Heterotic Line Bundle Models of Particle Physics with Genetic Algorithms and Quantum Annealing, Fortsch. Phys. 72 (2024) 2300260 [arXiv:2306.03147] [INSPIRE].
P. Berglund et al., New Calabi-Yau manifolds from genetic algorithms, Phys. Lett. B 850 (2024) 138504 [arXiv:2306.06159] [INSPIRE].
M. Cirafici, Persistent Homology and String Vacua, JHEP 03 (2016) 045 [arXiv:1512.01170] [INSPIRE].
A. Cole and G. Shiu, Persistent Homology and Non-Gaussianity, JCAP 03 (2018) 025 [arXiv:1712.08159] [INSPIRE].
A. Cole and G. Shiu, Topological Data Analysis for the String Landscape, JHEP 03 (2019) 054 [arXiv:1812.06960] [INSPIRE].
F. Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape, JHEP 08 (2017) 038 [arXiv:1706.07024] [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. Carifio, J. Halverson, D. Krioukov and B.D. Nelson, Machine Learning in the String Landscape, JHEP 09 (2017) 157 [arXiv:1707.00655] [INSPIRE].
A. Cole, A. Schachner and G. Shiu, Searching the Landscape of Flux Vacua with Genetic Algorithms, JHEP 11 (2019) 045 [arXiv:1907.10072] [INSPIRE].
J. Halverson and C. Long, Statistical Predictions in String Theory and Deep Generative Models, Fortsch. Phys. 68 (2020) 2000005 [arXiv:2001.00555] [INSPIRE].
H. Otsuka and K. Takemoto, Deep learning and k-means clustering in heterotic string vacua with line bundles, JHEP 05 (2020) 047 [arXiv:2003.11880] [INSPIRE].
R. Deen, Y.-H. He, S.-J. Lee and A. Lukas, Machine learning string standard models, Phys. Rev. D 105 (2022) 046001 [arXiv:2003.13339] [INSPIRE].
M. Bies et al., Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory, JHEP 01 (2021) 196 [arXiv:2007.00009] [INSPIRE].
A. Cole, S. Krippendorf, A. Schachner and G. Shiu, Probing the Structure of String Theory Vacua with Genetic Algorithms and Reinforcement Learning, in the proceedings of the 35th Conference on Neural Information Processing Systems, Online Conference, Canada, December 06–14 (2021) [arXiv:2111.11466] [INSPIRE].
G.J. Loges and G. Shiu, 134 billion intersecting brane models, JHEP 12 (2022) 097 [arXiv:2206.03506] [INSPIRE].
J. Halverson and F. Ruehle, Metric Flows with Neural Networks, arXiv:2310.19870 [INSPIRE].
S. Lanza, Machine learning the breakdown of tame effective theories, arXiv:2311.03437 [INSPIRE].
M. Dias, J. Frazer and A. Westphal, Inflation as an Information Bottleneck — A strategy for identifying universality classes and making robust predictions, JHEP 05 (2019) 065 [arXiv:1810.05199] [INSPIRE].
S.A. Abel, A. Constantin, T.R. Harvey and A. Lukas, Cosmic Inflation and Genetic Algorithms, Fortsch. Phys. 71 (2023) 2200161 [arXiv:2208.13804] [INSPIRE].
K. Hashimoto, S. Sugishita, A. Tanaka and A. Tomiya, Deep learning and the AdS/CFT correspondence, Phys. Rev. D 98 (2018) 046019 [arXiv:1802.08313] [INSPIRE].
K. Hashimoto, AdS/CFT correspondence as a deep Boltzmann machine, Phys. Rev. D 99 (2019) 106017 [arXiv:1903.04951] [INSPIRE].
T. Akutagawa, K. Hashimoto and T. Sumimoto, Deep Learning and AdS/QCD, Phys. Rev. D 102 (2020) 026020 [arXiv:2005.02636] [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, Annals Phys. 392 (2018) 93 [arXiv:1706.06430] [INSPIRE].
M.R. Douglas, The String Theory Landscape, Universe 5 (2019) 176 [INSPIRE].
F. Ruehle, Data science applications to string theory, Phys. Rept. 839 (2020) 1 [INSPIRE].
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B 766 (2007) 21 [hep-th/0605264] [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].
L. van den Dries, Tame topology and o-minimal structures, Cambridge University Press, Cambridge (1998) [https://doi.org/10.1017/cbo9780511525919].
T.W. Grimm, Taming the landscape of effective theories, JHEP 11 (2022) 003 [arXiv:2112.08383] [INSPIRE].
B. Bakker, T.W. Grimm, C. Schnell and J. Tsimerman, Finiteness for self-dual classes in integral variations of Hodge structure, arXiv:2112.06995 [https://doi.org/10.46298/epiga.2023.specialvolumeinhonourofclairevoisin.9626] [INSPIRE].
T.W. Grimm, S. Lanza and C. Li, Tameness, Strings, and the Distance Conjecture, JHEP 09 (2022) 149 [arXiv:2206.00697] [INSPIRE].
T.W. Grimm and J. Monnee, Finiteness Theorems and Counting Conjectures for the Flux Landscape, arXiv:2311.09295 [INSPIRE].
M.R. Douglas, T.W. Grimm and L. Schlechter, The Tameness of Quantum Field Theory, Part I – Amplitudes, arXiv:2210.10057 [INSPIRE].
M.R. Douglas, T.W. Grimm and L. Schlechter, The Tameness of Quantum Field Theory, Part II – Structures and CFTs, arXiv:2302.04275 [INSPIRE].
T.W. Grimm, L. Schlechter and M. van Vliet, Complexity in tame quantum theories, JHEP 05 (2024) 001 [arXiv:2310.01484] [INSPIRE].
M. Anthony and P.L. Bartlett, Neural Network Learning: Theoretical Foundations, Cambridge University Press (1999) [https://doi.org/10.1017/cbo9780511624216].
B.K. Natarajan, On learning sets and functions, Mach. Learn. 4 (2004) 67.
B. Bakker, B. Klingler and J. Tsimerman, Tame topology of arithmetic quotients and algebraicity of Hodge loci, J. Am. Math. Soc. 33 (2020) 917.
M.C. Laskowski, Vapnik-Chervonenkis Classes of Definable Sets, J. Lond. Math. Soc. s 2-45 (1992) 377.
M. Tressl, Introduction to o-minimal structures and an application to neural network learning, (2010) [https://api.semanticscholar.org/CorpusID:14543410].
D. Marker, Model Theory: An Introduction, Springer New York, (2002) [https://doi.org/10.1007/b98860].
I. Moerdijk and J. van Oosten, Sets, Models and Proofs, Springer International Publishing (2018) [https://doi.org/10.1007/978-3-319-92414-4].
A. Tarski, A decision method for elementary algebra and geometry, in Quantifier Elimination and Cylindrical Algebraic Decomposition, B.F. Caviness and J.R. Johnson, eds., Springer Vienna (1998), pp. 24–84 [https://doi.org/10.1007/978-3-7091-9459-1_3].
A. Seidenberg, A new decision method for elementary algebra, Annals Math. 60 (1954) 365.
L. Van Den Dries, Alfred Tarski’s elimination theory for real closed fields, J. Symb. Log. 53 (1988) 7.
A. Macintyre and A.J. Wilkie, On the decidability of the real exponential field, in Kreiseliana. About and Around Georg Kreisel, P. Odifreddi ed., A K Peters (1996), p. 441–467 [ISBN: 9781568810614].
Acknowledgments
I am deeply grateful to Thomas Grimm, Luca Martucci, Mick van Vliet, Timo Weigand, and AlexanderWestphal for valuable discussions and comments on the manuscript. This research is supported in part by Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 Quantum Universe 390833306 and by Deutsche Forschungsgemeinschaft through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”.
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Lanza, S. Neural network learning and Quantum Gravity. J. High Energ. Phys. 2024, 105 (2024). https://doi.org/10.1007/JHEP07(2024)105
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DOI: https://doi.org/10.1007/JHEP07(2024)105