Machine Learning Gravity Compactifications on Negatively Curved Manifolds

TL;DR

This work tackles the challenge of constructing vacua in higher-dimensional gravity by directly solving the non-linear Einstein equations for warped compactifications using neural networks. The authors formulate a physics-informed ML framework that parameterizes the internal geometry with neural nets, enforces the equations of motion and boundary/junction conditions via a tailored loss, and optimizes over network weights to obtain approximate solutions on patchwise manifolds. They demonstrate a proof-of-concept in three dimensions by building and filling a cusped hyperbolic manifold, achieving order-$10^{-3}$ accuracy in both the Einstein equations and boundary constraints, and they outline clear paths to extending the method to higher dimensions and to de Sitter compactifications in M-theory. The results indicate a promising, scalable route to numerically construct negatively curved Einstein metrics and, potentially, dS$_4$ vacua, with public code available to reproduce and extend the analysis.

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.

Figures (5)

• Figure 1: An embedding of $P_3$ in the UHS model \ref{['eq:UHS']} with $z$ the vertical coordinate. This polytope is composed of 6 totally geodesics facets: four vertical walls and two hemispheres centered at $z = 0$. In this picture the coordinate $z$ has been cut-off at a finite volume, but the polytope extends up to $z = \infty$, while remaining of finite volume with respect to the metric \ref{['eq:UHS']}.
• Figure 2: A view from the top of the larger polytope $P_3^8$ obtained by reflecting according to the reflections \ref{['eq:refl']}
• Figure 3: A typical training run. Left: pre-training loss when training to approximate the piece-wise solution starting from a random initialization, right: training loss (running average) when training on the full loss starting from the pre-trained network. The pre-training is early-stopped when the pre-training loss reaches a certain threshold, as its role is only to serve as a good initialization for the actual training run.
• Figure 4: Percentage deviation of the Ricci scalar from its true value for a $k = 5$ filling, evaluated on the slice $x_1 = x_2 = \frac{1}{2}$, after the pre-training phase (green) and after the training phase of a typical training run (purple), compared to two different manual smooth interpolations of the piece-wise Einstein metric. Shaded area shows 5% error threshold.
• Figure 5: Local Relative Error for all the components of the Einstein equation on a typical training run. Shaded area shows 5% error.