Search for Z/2 eigenfunctions on the sphere using machine learning
This work addresses a mathematical problem in geometry and analysis, providing incremental computational examples for a specialized domain.
The researchers tackled the problem of finding Z/2 eigenfunctions on the 2-sphere by developing a multivalued feedforward deep neural network using JAX, resulting in the discovery of such eigenfunctions for three specific geometric configurations, including fixed branch points at tetrahedron and cube vertices and a squashed tetrahedron from AI-optimized positioning.
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.