Predicting Euler Characteristics Using Machine Learning and Skyrmion Number Computation

Our study investigates the method to obtain topological properties of input images with neural networks, not requiring training datasets. In the field of solid-state physics, research has been conducted to obtain topological properties of magnetic structures by analyzing the spin fields. Utilizing the approaches, our model generates a unit vector field interpreted as spin fields from various images and predicts the Euler characteristic of input images by computing the skyrmion number of the generated vector field. Even if the networks are trained by a single image of a fixed Euler characteristic, they successfully predict the Euler characteristics of the various images. The resulting spin configurations from independently trained neural networks are not unique due to the remaining degrees of freedom in the spin configuration. To further control the spin configuration by confining these degrees of freedom, we incorporate a magnetic Hamiltonian as an additional loss function, which includes exchange Interaction, Dzyaloshinskii-Moriya (DM) Interaction, and anisotropy. We validate the model on more complex geometrical shapes and apply it to practical tasks.