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

• URL: http://arxiv.org/abs/2212.05892v1
• Date: Wed, 30 Nov 2022 08:17:06 GMT
• Title: A physics-informed search for metric solutions to Ricci flow, their embeddings, and visualisation
• Authors: Aarjav Jain, Challenger Mishra, Pietro Li\`o
• Abstract summary: Neural networks with PDEs embedded in their loss functions are employed as a function approximators. 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathbb{R}^3$. 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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