A physics-informed search for metric solutions to Ricci flow, their embeddings, and visualisation
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.
Cite
@article{arxiv.2212.05892,
title = {A physics-informed search for metric solutions to Ricci flow, their embeddings, and visualisation},
author = {Aarjav Jain and Challenger Mishra and Pietro Liò},
journal= {arXiv preprint arXiv:2212.05892},
year = {2022}
}