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

TL;DR

This paper introduces a physics-informed neural network approach to solving Ricci flow equations, demonstrating its effectiveness on the torus and exploring applications to Calabi–Yau metrics in string theory.

Contribution

It develops a general method using physics-informed neural networks for Ricci flow solutions and visualizations, with potential applications in complex geometry and string theory.

Findings

Neural network solutions match standard PDE solver results.

Visualizations of Ricci flow via embeddings into 3D.

Guidelines for applying the method to Ricci-flat Calabi–Yau metrics.

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.