Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties

TL;DR

This paper introduces a machine learning-based method to derive analytic formulas for line bundle cohomologies on hypersurfaces in toric varieties, overcoming limitations of naive neural network approaches.

Contribution

It presents a novel unsupervised learning approach that identifies piecewise polynomial structures, enabling explicit formula derivation for cohomologies.

Findings

Unsupervised learning separates polynomial phases in cohomology data.

The method yields explicit analytic formulas for cohomologies.

Algorithm can compute cohomologies for arbitrary hypersurfaces in toric varieties.

Abstract

Different techniques from machine learning are applied to the problem of computing line bundle cohomologies of (hypersurfaces in) toric varieties. While a naive approach of training a neural network to reproduce the cohomologies fails in the general case, by inspecting the underlying functional form of the data we propose a second approach. The cohomologies depend in a piecewise polynomial way on the line bundle charges. We use unsupervised learning to separate the different polynomial phases. The result is an analytic formula for the cohomologies. This can be turned into an algorithm for computing analytic expressions for arbitrary (hypersurfaces in) toric varieties.