Search for Z/2 eigenfunctions on the sphere using machine learning

TL;DR

This paper employs machine learning, specifically a multivalued neural network implemented in JAX, to discover Z/2 eigenfunctions on the 2-sphere, including cases with fixed and movable branch points.

Contribution

It introduces a multivalued neural network approach to find Z/2 eigenfunctions on the sphere, exploring configurations with fixed and adaptable branch points.

Findings

Found Z/2 eigenfunctions with fixed branch points at tetrahedron and cube vertices.

Discovered a configuration with movable branch points ending at a squashed tetrahedron.

Demonstrated the effectiveness of machine learning in geometric eigenfunction discovery.

Abstract

We use machine learning to search for examples of Z/2 eigenfunctions on the 2-sphere. For this we created a multivalued version of a feedforward deep neural network, and we implemented it using the JAX library. We found Z/2 eigenfunctions for three cases: In the first two cases we fixed the branch points at the vertices of a tetrahedron and at a cube respectively. In a third case, we allowed the AI to move the branch points around and, in the end, it positioned the branch points at the vertices of a squashed tetrahedron.