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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