Predicting Euler Characteristics and Constructing Topological Structure Using Machine Learning Techniques

TL;DR

This paper introduces a neural network approach to predict Euler characteristics from images by generating spin configurations and computing skyrmion numbers, incorporating physics-informed loss functions for improved accuracy.

Contribution

The method uniquely predicts topological properties from a single image without large datasets, integrating physics-based constraints to refine spin configuration generation.

Findings

Successfully predicts Euler characteristics for complex shapes.

Generates physically consistent spin configurations using physics-informed loss.

Applicable to practical topological analysis tasks.

Abstract

This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre-existing datasets but with a single geometric image. Inspired by solid-state physics, where topological properties of magnetic structures are derived from spin field analysis, our model generates a unit vector field from an image, interpreted as a spin configuration. The Euler characteristic is then predicted by computing the skyrmion number of this generated spin configuration. Remarkably, the network learns to construct chiral magnetic textures without access to ground-truth chiral spin configurations, relying instead on only a single, simple geometric image and the straightforward skyrmion number computation. Furthermore, spin configurations generated by independently trained networks can be non-unique due to inherent degrees of freedom. To constrain these degrees of freedom and further refine the spin configuration, we incorporate a magnetic Hamiltonian, comprising exchange interaction, Dzyaloshinskii-Moriya (DM) interaction, and anisotropy, as an additional, physics-informed loss function. We validate the model's efficacy on complex geometrical shapes and demonstrate its applicability to practical tasks.