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Rigor with machine learning from field theory to the Poincaré conjecture

Nature Reviews Physics (2024)Cite this article

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Abstract

Despite their successes, machine learning techniques are often stochastic, error-prone and blackbox. How could they then be used in fields such as theoretical physics and pure mathematics for which error-free results and deep understanding are a must? In this Perspective, we discuss techniques for obtaining zero-error results with machine learning, with a focus on theoretical physics and pure mathematics. Non-rigorous methods can enable 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 4D Poincaré conjecture in low-dimensional topology. We also discuss connections between machine learning theory and mathematics or theoretical physics such as 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 used to solve the 3D Poincaré conjecture.

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Fig. 1: Moves and singularities of ribbon knots.
Fig. 2: Performance comparison of the TRPO, A3C and RW algorithms on the unknotting problem.

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Acknowledgements

S.G. is supported in part by a Simons Collaboration Grant on New Structures in Low-Dimensional Topology and by the Department of Energy grant DE-SC0011632. J.H. is supported by National Science Foundation CAREER grant PHY-1848089. F.R. is supported by the National Science Foundation grants PHY-2210333 and startup funding from Northeastern University. The work of J.H. and F.R. is also supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions).

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Authors and Affiliations

  1. Richard N. Merkin Center for Pure and Applied Mathematics, California Institute of Technology, Pasadena, CA, USA

    Sergei Gukov

  2. Dublin Institute for Advanced Studies, Dublin, Ireland

    Sergei Gukov

  3. Department of Physics, Northeastern University, Boston, MA, USA

    James Halverson & Fabian Ruehle

  4. The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, Cambridge, USA

    James Halverson & Fabian Ruehle

  5. Department of Mathematics, Northeastern University, Boston, MA, USA

    Fabian Ruehle

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Gukov, S., Halverson, J. & Ruehle, F. Rigor with machine learning from field theory to the Poincaré conjecture. Nat Rev Phys (2024). https://doi.org/10.1038/s42254-024-00709-0

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