Paper · Feb 20, 2024

Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

Sergei Gukov, James Halverson, Fabian Ruehle

hep-tharXiv:2402.13321
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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 d Poincaré 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 d Poincaré conjecture.

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Submitted
Feb 20, 2024
Subjects
hep-th, cs.LG

Comments: 13 pages. Preprint of edited version in Nature Reviews Physics. Please cite journal version

Ricci flowPoincaré conjecturePhysicsDiscrete & graph Ricci

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Ricci Flow and the Poincare Conjecture (John W. Morgan and Gang Tian, 2006)3Morgan 2006 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)7Cao 2006 Dynamically Stable Poincaré Embeddings for Neural Manifolds (Jun Chen, Yuang Liu, Xiangrui Zhao et al., 2021)4Chen 2021 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)2Akbulut 2022 Metric Flows with Neural Networks (James Halverson and Fabian Ruehle, 2023)1Halverson 2023 Moving Manifolds and the Poincare Conjecture (David V. Svintradze, 2025)8Svintradze 2025 Equations in Products of Free Groups and 3-Manifold Groups, I (Olga Kharlampovich and Alina Vdovina, 2026)5Kharlampovic 2026 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)6Qin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Metric Flows with Neural Networks

    James Halverson and Fabian Ruehle

    Topics Poincaré conjecture, Discrete & graph Ricci · Same authors James Halverson, Fabian Ruehle · Shared terms “neural networks”, “learning”, “gradient descent”

  2. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut

    Topics Poincaré conjecture

  3. Ricci Flow and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon

    Topics Poincaré conjecture · Shared terms “Perelman”

  4. Dynamically Stable Poincaré Embeddings for Neural Manifolds

    Jun Chen, Yuang Liu, Xiangrui Zhao et al.

    Topics Discrete & graph Ricci · Shared terms “machine learning”, “neural networks”, “Poincaré”

  5. Equations in Products of Free Groups and 3-Manifold Groups, I

    Olga Kharlampovich and Alina Vdovina

    Topics Poincaré conjecture · Shared terms “dimensional topology”, “low dimensional”, “Perelman”

  6. A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

    Ziyang Qin, Yuan Liao, Ayush Khaitan et al.

    Topics Poincaré conjecture · Shared terms “Perelman”

  7. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

    Huai-Dong Cao and Xi-Ping ZhuThe canon

    Topics Poincaré conjecture · Shared terms “breakthrough”, “Perelman”

  8. Moving Manifolds and the Poincare Conjecture

    David V. Svintradze

    Topics Poincaré conjecture · Shared terms “formulation”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Poincaré conjecture Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré posed the question in 1904, and Perelman proved it in 2002–03… Start with: Generalized Poincare's Conjecture in Dimensions Greater Than Four (Smale, 1961) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  2. 7A.12
    The Poincaré Conjecture, PreciselyBook 7A · Topology and the Fundamental Group

    Every word of the statement, and why each hypothesis is needed.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.