Machine Learning Gravity Compactifications on Negatively Curved Manifolds

TL;DR

This paper explores using Machine Learning, specifically Neural Networks, to numerically solve Einstein equations for complex warped gravity compactifications on negatively curved manifolds, aiming to find new Einstein metrics and de Sitter vacua.

Contribution

It demonstrates the feasibility of applying Neural Networks to solve Einstein PDEs on hyperbolic manifolds, enabling the construction of higher-dimensional negatively curved Einstein metrics.

Findings

Neural Networks successfully solve Einstein PDEs on non-trivial hyperbolic manifolds.

The methods are scalable and applicable to higher dimensions.

Potential to construct novel Einstein metrics and de Sitter vacua in M-theory.

Abstract

Constructing the landscape of vacua of higher-dimensional theories of gravity by directly solving the low-energy (semi-)classical equations of motion is notoriously difficult. In this work, we investigate the feasibility of Machine Learning techniques as tools for solving the equations of motion for general warped gravity compactifications. As a proof-of-concept we use Neural Networks to solve the Einstein PDEs on non-trivial three manifolds obtained by filling one or more cusps of hyperbolic manifolds. While in three dimensions an Einstein metric is also locally hyperbolic, the generality and scalability of Machine Learning methods, the availability of explicit families of hyperbolic manifolds in higher dimensions, and the universality of the filling procedure strongly suggest that the methods and code developed in this work can be of broader applicability. Specifically, they can be used to tackle both the geometric problem of numerically constructing novel higher-dimensional negatively curved Einstein metrics, as well as the physical problem of constructing four-dimensional de Sitter compactifications of M-theory on the same manifolds.