Machine learning used to probe the building blocks of shapes

Peer-Reviewed Publication

IMPERIAL COLLEGE LONDON

 

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THIS IMAGE SHOWS A SLICE OF THE FANO VARIETY V6.

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CREDIT: IMPERIAL COLLEGE LONDON

Applying machine learning to find the properties of atomic pieces of geometry shows how AI has the power to accelerate discoveries in maths.

Mathematicians from Imperial College London and the University of Nottingham have, for the first time, used machine learning to expand and accelerate work identifying ‘atomic shapes’ that form the basic pieces of geometry in higher dimensions. Their findings have been published in Nature Communications.

The way they used artificial intelligence, in the form of machine learning, could transform how maths is done, say the authors. Dr Alexander Kasprzyk from the University of Nottingham said: “For mathematicians, the key step is working out what the pattern is in a given problem. This can be very difficult, and some mathematical theories can take years to discover.”

Professor Tom Coates, from the Department of Mathematics at Imperial, added: “We have shown that machine learning can help uncover patterns within mathematical data, giving us both new insights and hints of how they can be proved.”

PhD student Sara Veneziale, from the Department of Mathematics at Imperial, said: “This could be very broadly applicable, such that it could rapidly accelerate the pace at which maths discoveries are made. It’s like when computers were first used in maths research, or even calculators: it’s a step-change in the way we do maths.”

Defining shapes

Mathematicians describe shapes using equations, and by analysing these equations can break the shape down into fundamental pieces. These are the building blocks of shapes, the equivalent of atoms, and are called Fano varieties.

The Imperial and Nottingham team began building a ‘periodic table’ of these Fano varieties several years ago, but finding ways of classifying them into groups with common properties has been challenging. Now, they have used machine learning to reveal unexpected patterns in the Fano varieties.

One aspect of a Fano variety is its quantum period – a sequence of numbers that acts like a barcode or fingerprint. It has been suggested that the quantum period defines the dimension of the Fano variety, but there has been no theoretical proposal for how this works, so no way to test it on the huge set of known Fano varieties.

Machine learning, however, is built to find patterns in large datasets. By training a machine learning model with some example data, the team were able to show the resulting model could predict the dimensions of Fano varieties from quantum periods with 99% accuracy.

Coding the real world

The AI model doesn’t conclusively show the team have discovered a new statement, so they then used more traditional mathematical methods to prove that the quantum period defines the dimension – using the AI model to guide them.

As well as using machine learning to discover new maths, the team say that the datasets used in maths could help refine machine learning models. Most models are trained on data taken from real life, such as health or transport data, which are inherently ‘noisy’ – they contain a lot of randomness that to some degree mask the real information.

Mathematical data is ‘pure’ – noise free – and contains patterns and structures that underly the data, waiting to be uncovered. This data can therefore be used as testing grounds for machine learning models, improving their ability to find new patterns.

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JOURNAL

Nature Communications

DOI

10.1038/s41467-023-41157-1 

ARTICLE TITLE

Machine learning the dimension of a Fano variety