Abstract
A team of scientists from School of Physics and Astronomy at Queen Mary University of London has developed a novel artificial intelligence method that could revolutionize our understanding of the universe's most mysterious shapes. Using advanced machine learning, researchers can now explore complex geometric spaces, like the fabric of spacetime itself, without relying on traditional symmetry assumptions.
This new algorithm, called AInstein, tackles one of the most complex puzzles in physics and mathematics: finding the precise shape of space under Einstein field equations. Remarkably, it can do so on spaces as intricate as higher-dimensional spheres, opening new avenues for discovery and shedding light on our understanding of the universe.
Introduction
This work introduces a novel semi-supervised machine learning approach, AInstein, designed to approximate Einstein metrics on a broad class of manifolds without relying on symmetry assumptions. Einstein metrics are fundamental in both mathematics and theoretical physics, as they describe the geometry of space in solutions to Einstein's equations, which are the base of our understanding of gravity, black holes, and the structure of the universe. Traditionally, finding these metrics, especially in higher dimensions or complex topologies, has been a formidable challenge, often requiring computationally intensive numerical methods and assumptions of symmetry.
The newly introduced package employs a ‘network-of-networks’ architecture to represent curved manifolds as collections of overlapping coordinate patches, a construction well known in geometry. Each subnetwork predicts the local metric — the mathematical object that encodes distances and curvature — within its patch. The Einstein field equations,
Rµν − λ gµν = 0,
are then incorporated as loss components that guide the optimisation process. These equations, first formulated by Albert Einstein in 1915, describe how the curvature of space and time is shaped by energy and matter. In the model, the Einstein constant λ can take values corresponding to positive, zero, or negative curvature, allowing the network to learn a broad class of possible geometries.
To ensure that the learned geometry is consistent across the entire space, the model in- troduces an additional loss term that enforces the correct transformation of the metric between neighbouring patches. This uses the tensorial relation g′ = J T gJ for a known Jacobian J , ensuring that the locally learned metrics join smoothly into a coherent global structure. This patch-based consistency requirement is what allows the method to con- struct valid manifolds without assuming any particular symmetry or coordinate system in advance.
The researchers tested the framework on spheres in dimensions two through five, each represented by two overlapping patches. In two and three dimensions, the geometry of the sphere is completely understood and serves as a useful benchmark. However, in four and five dimensions, it is not known whether a Ricci-flat metric — one with zero curvature in the Einstein sense — can exist on a sphere. The team’s results hint that such metrics are unlikely to exist, offering a computational indication that supports long- standing conjectures in differential geometry.
Beyond confirming known results, the framework demonstrates that deep-learning ar- chitectures can capture globally consistent geometric structures that satisfy nonlinear differential equations of physical significance. It shows how machine learning can be used not just for pattern recognition or data analysis, but as a theoretical tool for probing mathematical spaces that resist direct analytic treatment.
‘This is the first demonstration that neural networks can learn globally con- sistent solutions to Einstein’s equations without presupposing symmetry,’ the authors said in a joint statement.‘It opens up a new computational avenue for exploring geometries that are analytically intractable and may help uncover new classes of solutions that have so far remained hidden.’
The approach could open the door to studying a wider range of geometric spaces — including exotic spheres, black-hole geometries, and other manifolds with non- trivial curvature — where analytical methods are difficult to apply. By combining rigorous geometric reasoning with flexible machine-learning tools, the Queen Mary team has created a framework that bridges modern AI with some of the most intricate questions in pure and mathematical physics.
About the Authors
Edward Hirst — Hirst’s interests lie at the intersection of geometry and artificial intelligence, exploring how machine-learning methods can uncover new geometric struc- tures and relationships among solutions of Einstein’s equations.
Tancredi Schettini Gherardini — Schettini Gherardini focuses on geometry and exotic mathematical structures, developing AI frameworks that probe unconventional manifolds and geometries.
Alexander G. Stapleton — Stapleton works on information geometry, AI, renor- malisation, and the numerical conformal bootstrap, seeking to connect modern computational methods with deep questions in mathematical physics and quantum field theory.
Reference
Edward Hirst, Tancredi Schettini Gherardini and Alexander G Stapleton. AInstein: numerical Einstein metrics via machine learning[J]. AI For Science, 2025, 4(4): 042101. DOI: 10.1088/3050-287X/ae1117