Automatic classification of magnetic field lines by persistent homology

TL;DR

This paper introduces a novel method using persistent homology to automatically classify magnetic field line orbits into topologically distinct classes, aiding in understanding magnetic confinement in fusion devices.

Contribution

The paper presents a new topological data analysis approach using persistent homology for classifying magnetic field line orbits, improving accuracy over traditional methods.

Findings

Persistent H1 data distinguishes magnetic islands from other orbits.

Combining H0 and H1 data separates chaotic layers from invariant tori.

Coordinate transformation enhances classification performance.

Abstract

A method for the automatic classification of the orbits of magnetic field lines into topologically distinct classes using the Vietoris-Rips persistent homology is presented. The input to the method is the Poincare map orbits of field lines and the output is a separation into three classes: islands, chaotic layers, and invariant tori. The classification is tested numerically for the case of a toy model of a perturbed tokamak represented initially in its geometric coordinates. The persistent $H_1$ data is demonstrated to be sufficient to distinguish magnetic islands from the other orbits. When combined with persistent $H_0$ information, describing the average spacing between points on the Poincare section, the larger chaotic orbits can then be separated from very thin chaotic layers and invariant tori. It is then shown that if straight field line coordinates exist for a nearby integrable field configuration, the performance of the classification can be improved by transforming into this natural coordinate system. The focus is the application to toroidal magnetic confinement but the method is sufficiently general to apply to generic $1\frac{1}{2}$d Hamiltonian systems.