Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
166 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Rigor with Machine Learning from Field Theory to the Poincaré Conjecture (2402.13321v1)

Published 20 Feb 2024 in hep-th and cs.LG

Abstract: Machine learning techniques are increasingly powerful, leading to many breakthroughs in the natural sciences, but they are often stochastic, error-prone, and blackbox. How, then, should they be utilized in fields such as theoretical physics and pure mathematics that place a premium on rigor and understanding? In this Perspective we discuss techniques for obtaining rigor in the natural sciences with machine learning. Non-rigorous methods may lead to rigorous results via conjecture generation or verification by reinforcement learning. We survey applications of these techniques-for-rigor ranging from string theory to the smooth $4$d Poincar\'e conjecture in low-dimensional topology. One can also imagine building direct bridges between machine learning theory and either mathematics or theoretical physics. As examples, we describe a new approach to field theory motivated by neural network theory, and a theory of Riemannian metric flows induced by neural network gradient descent, which encompasses Perelman's formulation of the Ricci flow that was utilized to resolve the $3$d Poincar\'e conjecture.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (64)
  1. J. Jumper, R. Evans, A. Pritzel, T. Green, M. Figurnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. Zidek, A. Potapenko, A. Bridgland, C. Meyer, S. A. A. Kohl, A. J. Ballard, A. Cowie, B. Romera-Paredes, S. Nikolov, R. Jain, J. Adler, T. Back, S. Petersen, D. Reiman, E. Clancy, M. Zielinski, M. Steinegger, M. Pacholska, T. Berghammer, S. Bodenstein, D. Silver, O. Vinyals, A. W. Senior, K. Kavukcuoglu, P. Kohli, and D. Hassabis “Highly accurate protein structure prediction with alphafold,” Nature 596 (Aug, 2021) 583–589.
  2. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová “Machine learning and the physical sciences,” Rev. Mod. Phys. 91 (2019) no. 4, 045002 [1903.10563].
  3. F. Ruehle “Data science applications to string theory,” Phys. Rept. 839 (2020) 1–117.
  4. Y. He Machine Learning in Pure Mathematics and Theoretical Physics. G - Reference,Information and Interdisciplinary Subjects Series. World Scientific 2023.
  5. A. Athalye, L. Engstrom, A. Ilyas, and K. Kwok “Synthesizing robust adversarial examples,” PMLR 80 (2018) [1707.07397].
  6. A. Athalye, N. Carlini, and D. Wagner “Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples,” in Proceedings of the 35th International Conference on Machine Learning, ICML 2018. July, 2018.
  7. A. Jacot, F. Gabriel, and C. Hongler “Neural tangent kernel: Convergence and generalization in neural networks,” in NeurIPS. 2018.
  8. J. Lee, L. Xiao, S. S. Schoenholz, Y. Bahri, R. Novak, J. Sohl-Dickstein, and J. Pennington “Wide neural networks of any depth evolve as linear models under gradient descent,” ArXiv abs/1902.06720 (2019).
  9. R. S. Hamilton “Three-manifolds with positive Ricci curvature,” Journal of Differential Geometry 17 (1982) no. 2, 255 – 306.
  10. G. Perelman “The entropy formula for the ricci flow and its geometric applications,” 2002.
  11. T. C. Hales “Developments in formal proofs,” Astérisque (2015) no. 367-368, Exp. No. 1086, x, 387–410.
  12. J. Alama, T. Heskes, D. Kühlwein, E. Tsivtsivadze, and J. Urban “Premise selection for mathematics by corpus analysis and kernel methods,” J. Automat. Reason. 52 (2014) no. 2, 191–213.
  13. J. C. Blanchette, D. Greenaway, C. Kaliszyk, D. Kühlwein, and J. Urban “A learning-based fact selector for Isabelle/HOL,” J. Automat. Reason. 57 (2016) no. 3, 219–244.
  14. Y. Nagashima “Simple dataset for proof method recommendation in isabelle/hol (dataset description),” arXiv (2020) [2004.10667].
  15. B. Piotrowski, R. F. Mir, and E. Ayers “Machine-learned premise selection for lean,” arXiv (2023) [2304.00994].
  16. J. Carifio, J. Halverson, D. Krioukov, and B. D. Nelson “Machine Learning in the String Landscape,” JHEP 09 (2017) 157 [1707.00655].
  17. Y.-H. He “Deep-Learning the Landscape,” arXiv (6, 2017) [1706.02714].
  18. D. Krefl and R.-K. Seong “Machine Learning of Calabi-Yau Volumes,” Phys. Rev. D 96 (2017) no. 6, 066014 [1706.03346].
  19. F. Ruehle “Evolving neural networks with genetic algorithms to study the String Landscape,” JHEP 08 (2017) 038 [1706.07024].
  20. A. Davies, P. Veličković, L. Buesing, S. Blackwell, D. Zheng, N. Tomašev, R. Tanburn, P. Battaglia, C. Blundell, A. Juhász, M. Lackenby, G. Williamson, D. Hassabis, and P. Kohli “Advancing mathematics by guiding human intuition with ai,” Nature 600 (Dec, 2021) 70–74.
  21. J. Craven, V. Jejjala, and A. Kar “Disentangling a deep learned volume formula,” JHEP 06 (2021) 040 [2012.03955].
  22. J. Craven, M. Hughes, V. Jejjala, and A. Kar “Learning knot invariants across dimensions,” SciPost Phys. 14 (2023) no. 2, Paper No. 021, 28.
  23. G. Brown, T. Coates, A. Corti, T. Ducat, L. Heuberger, and A. Kasprzyk “Computation and data in the classification of fano varieties,” 2022.
  24. S. Gukov and R.-K. Seong “Learning BPS spectra, to appear,” 2023.
  25. C. Mishra, S. R. Moulik, and R. Sarkar “Mathematical conjecture generation using machine intelligence,” 2023.
  26. D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. van den Driessche, T. Graepel, and D. Hassabis “Mastering the game of go without human knowledge,” Nature 550 (Oct, 2017) 354–359.
  27. D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, T. Lillicrap, K. Simonyan, and D. Hassabis “Mastering chess and shogi by self-play with a general reinforcement learning algorithm,” 2017.
  28. D. Klaewer and L. Schlechter “Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties,” Phys. Lett. B 789 (2019) 438–443 [1809.02547].
  29. C. R. Brodie, A. Constantin, R. Deen, and A. Lukas “Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces,” Compl. Manif. 8 (2021) no. 1, 223–229 [1906.08363].
  30. C. R. Brodie, A. Constantin, R. Deen, and A. Lukas “Index Formulae for Line Bundle Cohomology on Complex Surfaces,” Fortsch. Phys. 68 (2020) no. 2, 1900086 [1906.08769].
  31. C. R. Brodie, A. Constantin, R. Deen, and A. Lukas “Machine Learning Line Bundle Cohomology,” Fortsch. Phys. 68 (2020) no. 1, 1900087 [1906.08730].
  32. C. R. Brodie and A. Constantin “Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds,” arXiv (9, 2020) [2009.01275].
  33. M. Bies, M. Cvetič, R. Donagi, L. Lin, M. Liu, and F. Ruehle “Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory,” JHEP 01 (2021) 196 [2007.00009].
  34. M. C. Hughes “A neural network approach to predicting and computing knot invariants,” J. Knot Theory Ramifications 29 (2020) no. 3, 2050005, 20.
  35. S. Gukov, J. Halverson, C. Manolescu, and F. Ruehle “Searching for ribbons with machine learning,” arXiv (2023) [2304.09304].
  36. J. Hass, J. C. Lagarias, and N. Pippenger “The computational complexity of knot and link problems,” J. ACM 46 (1999) no. 2, 185–211.
  37. G. Kuperberg “Knottedness is in NP, modulo GRH,” Adv. Math. 256 (2014) 493–506.
  38. M. Lackenby “The efficient certification of knottedness and thurston norm,” Advances in Mathematics 387 (2021) 107796.
  39. S. Gukov, J. Halverson, F. Ruehle, and P. Sułkowski “Learning to unknot,” Machine Learning: Science and Technology 2 (apr, 2021) 025035.
  40. J. W. Alexander “A lemma on systems of knotted curves,” Proceedings of the National Academy of Sciences of the United States of America 9 (1923) no. 3, 93–95.
  41. S. J. Ri and P. Putrov “Graph Neural Networks and 3-dimensional topology,” Mach. Learn. Sci. Tech. 4 (2023) no. 3, 035026 [2305.05966].
  42. S. Gukov, J. Halverson, C. Manolescu, and F. Ruehle “An algorithm for finding ribbon bands.” ”https://github.com/ruehlef/ribbon” 2023.
  43. R. M. Neal BAYESIAN LEARNING FOR NEURAL NETWORKS. PhD thesis University of Toronto 1995.
  44. C. K. Williams “Computing with infinite networks,” in Advances in neural information processing systems pp. 295–301. 1997.
  45. G. Yang “Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes,” arXiv e-prints (Oct., 2019) arXiv:1910.12478 [1910.12478].
  46. J. Halverson, A. Maiti, and K. Stoner “Neural Networks and Quantum Field Theory,” Mach. Learn. Sci. Tech. 2 (2021) no. 3, 035002 [2008.08601].
  47. J. Halverson “Building quantum field theories out of neurons,” arXiv (2021) [2112.04527].
  48. M. Demirtas, J. Halverson, A. Maiti, M. D. Schwartz, and K. Stoner “Neural Network Field Theories: Non-Gaussianity, Actions, and Locality,” arXiv (7, 2023) [2307.03223].
  49. K. Osterwalder and R. Schrader “Axioms for euclidean green’s functions,” Communications in Mathematical Physics 31 (Jun, 1973) 83–112.
  50. Cambridge University Press 2022.
  51. A. Maiti, K. Stoner, and J. Halverson “Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators,” arXiv (6, 2021) [2106.00694].
  52. J. Halverson and F. Ruehle “Metric flows with neural networks,” 2023.
  53. S.-T. Yau “On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I,” Commun. Pure Appl. Math. 31 (1978) no. 3, 339–411.
  54. E. Calabi On Kähler Manifolds with Vanishing Canonical Class: pp. 78–89. Princeton University Press 2015.
  55. S. K. Donaldson “Some numerical results in complex differential geometry,” 2005.
  56. L. B. Anderson, M. Gerdes, J. Gray, S. Krippendorf, N. Raghuram, and F. Ruehle “Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning,” JHEP 05 (2021) 013 [2012.04656].
  57. M. R. Douglas, S. Lakshminarasimhan, and Y. Qi “Numerical Calabi-Yau metrics from holomorphic networks,” arXiv (12, 2020) [2012.04797].
  58. V. Jejjala, D. K. Mayorga Pena, and C. Mishra “Neural network approximations for Calabi-Yau metrics,” JHEP 08 (2022) 105 [2012.15821].
  59. M. Larfors, A. Lukas, F. Ruehle, and R. Schneider “Learning Size and Shape of Calabi-Yau Spaces,” in Fourth Workshop on Machine Learning and the Physical Sciences. 11, 2021. [2111.01436].
  60. M. Larfors, A. Lukas, F. Ruehle, and R. Schneider “Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds,” Mach. Learn. Sci. Tech. 3 (2022) no. 3, 035014 [2205.13408].
  61. M. Gerdes and S. Krippendorf “CYJAX: A package for Calabi-Yau metrics with JAX,” Mach. Learn. Sci. Tech. 4 (2023) no. 2, 025031 [2211.12520].
  62. G. Yang “Tensor programs ii: Neural tangent kernel for any architecture,” ArXiv abs/2006.14548 (2020).
  63. J. Cotler and S. Rezchikov “Renormalization group flow as optimal transport,” Physical Review D 108 (jul, 2023).
  64. D. S. Berman and M. S. Klinger “The Inverse of Exact Renormalization Group Flows as Statistical Inference,” arXiv (12, 2022) [2212.11379].
Citations (8)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now