Paper · Nov 30, 2022

A physics-informed search for metric solutions to Ricci flow, their embeddings, and visualisation

Aarjav Jain, Challenger Mishra, Pietro Liò

gr-qcarXiv:2212.05892
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Abstract

Neural networks with PDEs embedded in their loss functions (physics-informed neural networks) are employed as a function approximators to find solutions to the Ricci flow (a curvature based evolution) of Riemannian metrics. A general method is developed and applied to the real torus. The validity of the solution is verified by comparing the time evolution of scalar curvature with that found using a standard PDE solver, which decreases to a constant value of 0 on the whole manifold. We also consider certain solitonic solutions to the Ricci flow equation in two real dimensions. We create visualisations of the flow by utilising an embedding into . Snapshots of highly accurate numerical evolution of the toroidal metric over time are reported. We provide guidelines on applications of this methodology to the problem of determining Ricci flat Calabi–Yau metrics in the context of String theory, a long standing problem in complex geometry.

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Submitted
Nov 30, 2022
Subjects
gr-qc, cs.NE, math-ph
Ricci flowRicci-flat & Calabi–YauKähler geometryScalar curvaturePhysicsDiscrete & graph RicciNumerics & visualization

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 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)7Jiang 2021 Dynamically Stable Poincaré Embeddings for Neural Manifolds (Jun Chen, Yuang Liu, Xiangrui Zhao et al., 2021)2Chen 2021 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)8Conboye 2023 Metric Flows with Neural Networks (James Halverson and Fabian Ruehle, 2023)1Halverson 2023 Rigor with Machine Learning from Field Theory to the Poincaré Conjecture (Sergei Gukov, James Halverson and Fabian Ruehle, 2024)6Gukov 2024 Ricci Flow on Weighted Digraphs with Balancing Factor (Shuliang Bai, Rui Li, Shuang Liu et al., 2025)3Bai 2025 Neural Feature Geometry Evolves as Discrete Ricci Flow (Moritz Hehl, Max von Renesse and Melanie Weber, 2025)5Hehl 2025 Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry (Yeong-Bok Bae, Young-Hwan Hyun and Gungwon Kang, 2026)4Bae 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 Numerics & visualization, Discrete & graph Ricci, Ricci-flat & Calabi–Yau · Shared terms “neural networks”, “Yau metrics”, “numerical”

  2. Dynamically Stable Poincaré Embeddings for Neural Manifolds

    Jun Chen, Yuang Liu, Xiangrui Zhao et al.

    Topics Discrete & graph Ricci · Shared terms “neural networks”, “embedding”

  3. Ricci Flow on Weighted Digraphs with Balancing Factor

    Shuliang Bai, Rui Li, Shuang Liu et al.

    Topics Numerics & visualization, Discrete & graph Ricci · Shared terms “numerical”

  4. Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry

    Yeong-Bok Bae, Young-Hwan Hyun and Gungwon Kang

    Topics Numerics & visualization, Discrete & graph Ricci, Physics · Shared terms “embedding”, “numerical”

  5. Neural Feature Geometry Evolves as Discrete Ricci Flow

    Moritz Hehl, Max von Renesse and Melanie Weber

    Topics Discrete & graph Ricci · Shared terms “neural networks”, “informed”

  6. Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

    Sergei Gukov, James Halverson and Fabian Ruehle

    Topics Discrete & graph Ricci, Physics · Shared terms “neural networks”, “string theory”

  7. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang

    Topics Ricci-flat & Calabi–Yau, Scalar curvature

  8. Stability of piecewise flat Ricci flow

    Rory Conboye

    Topics Numerics & visualization · Shared terms “numerical”, “developed”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.