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DeepMind, the AI research laboratory funded by Google’s parent company, Alphabet, today published the results of a collaboration between it and mathematicians to apply AI toward discovering new insights in areas of mathematics. DeepMind claims that its AI technology helped to uncover a new formula for a previously-unsolved conjecture, as well as a connection between different areas of mathematics elucidated by studying the structure of knots.
DeepMind’s experiments with AI run the gamut from systems that can win at StarCraft II and Go to machine learning models for app recommendations and datacenter cooling optimization. But the sciences remain of principle interest to DeepMind, not least of which because of their commercial applications. Earlier this year, DeepMind cofounder Demis Hassabis announced the launch of Isomorphic Labs, which will use machine learning to identify disease treatments that have thus far eluded researchers. Separately, the lab has spotlighted its work in the fields of weather forecasting, materials modeling, and atomic energy computation.
“At DeepMind, we believe that AI techniques are already sufficient to have a foundational impact in accelerating scientific progress across many different disciplines,” DeepMind machine learning specialist Alex Davies said in a statement. “Pure maths is one example of such a discipline, and we hope that [our work] can inspire other researchers to consider the potential for AI as a useful tool in the field.”
Applying AI to mathematics
DeepMind isn’t the first to apply AI to mathematics, setting aside the fact that mathematics is the foundation of all AI systems.
In 2020, Microsoft-backed AI research lab OpenAI introduced GPT-f, an automated prover and proof assistant for the Metamath formalization language. (In mathematics, a “proof” refers to a logical argument that tries to show that a statement is true.) GPT-f found new proofs that were accepted into a mathematics community, which the researchers claimed at the time was a historic achievement.
More recently, a group of researchers from the Technion in Israel and Google presented an automated conjecturing system called the Ramanujan Machine, which came up with original formulas for universal constants that show up in mathematics. One of the formulas created by the machine can be used to compute the value of a constant called Catalan’s number more efficiently than any human-discovered formula.
What ostensibly sets DeepMind’s work apart, however, is its detection of the existence of patterns in mathematics with supervised learning — and giving insight into these patterns with attribution techniques from AI. Supervised learning is defined by its use of labeled datasets to train algorithms to classify data, predict outcomes, and more, and it’s been applied to domains including fraud detection, sales forecasting, and inventory optimization.
In a paper published in the journal Nature, DeepMind describes how it — alongside professor Geordie Williamson at the University of Sydney — used AI to help discover a new approach to a longstanding conjecture in representation theory. Defying progress for nearly 40 years, the combinatorial invariance conjecture states that a relationship should exist between certain directed graphs and polynomials. (A directed graph is a set of vertices connected by edges, with each node having a direction associated with it.) Using machine learning techniques, DeepMind was able to gain confidence that such a relationship does indeed exist and to hypothesize that it might be related to structures known as “broken dihedral intervals” and “external reflections.” With this knowledge, professor Williamson was able to create an algorithm that would solve the combinatorial invariance conjecture, which DeepMind computationally verified across more than 3 million examples.
“One might imagine that the work of a mathematician is dry and formulaic. The reality is completely different. Mathematicians inhabit a world rich in imagination, heuristics, and intuition,” Williamson said in a statement. “Often finding the right way to think about something, even if imprecise, is more useful than another long calculation. It has been a fascinating interdisciplinary journey with the teams at DeepMind and Oxford. We have seen that machine learning can be used to guide intuition, and eventually to prove new theorems.”
The paper also details DeepMind’s work with professor Marc Lackenby and professor András Juhász at the University of Oxford, which explored knots — one of the fundamental objects of study in topology (i.e., the mathematical study of the properties that are preserved through deformations, twistings, and stretchings). An AI system trained by DeepMind revealed that a particular algebraic quantity — the “signature” — was directly related to the geometry of a knot, which wasn’t previously known or suggested by an existing theory. The lab guided professor Lackenby to discover a new quantity — “natural slope” — and prove the exact nature of the relationship by using attribution techniques from machine learning, establishing connections between different branches of mathematics.
As DeepMind notes, knots not only show the many ways a rope can be tangled, but also have connections with quantum field theory and non-Euclidean geometry. Algebra, geometry, and quantum theory all share unique perspectives on these objects, and a longstanding mystery is how these different branches relate.
Promise for discovery
DeepMind believes that the Nature paper, along with yet-to-be-released companion papers for each result, demonstrate the usefulness of machine learning as a tool for mathematical study. AI excels at identifying and discovering patterns in data, the lab asserts, even exceeding the capabilities of expert human mathematicians.
“Finding patterns has become even more important in pure mathematics because it’s now possible to generate more data than any mathematician can reasonably expect to study in a lifetime. Some objects of interest — such as those with thousands of dimensions — can also simply be too unfathomable to reason about directly. With these constraints in mind, we believed that AI would be capable of augmenting mathematicians’ insights in entirely new ways,” DeepMind wrote in a blog post.
Queen Mary University professor of computational creativity Simon Colton, who wasn’t involved in the research, said that this is likely the first time deep learning techniques have been used for mathematical discovery. But he questioned whether mathematicians would want machine learning systems to take the creative lead in projects.
“When I was working with mathematicians, it was clear that they were happy for AI systems to prove minor things like lemmas and side conditions, etc., and to do huge calculations as per computer algebra systems. However, they were not happy for an AI system to prove important results (especially if they couldn’t understand the proof), or to perform concept invention, as this was the creative part of the job they loved the most,” Colton told VentureBeat via email. “With notable exceptions, the vast majority of theorems in pure mathematics are as useful to society as a painting by an amateur, i.e., only of interest to a small clique of people. So, it’s not safety-critical to the progression of society or general well-being to have AI systems involved in pure mathematics (like it is for protein folding, another area that DeepMind has innovated in).”
Still, Colton expects that the broader adoption of AI systems in pure mathematics — assuming it occurs — will lead to interesting discoveries “that may be beyond human comprehension.”
“We may therefore find a limit on what mathematicians can verify and what they want AI systems to do,” he continued. “It’s great that DeepMind [is] getting into this area and working with top mathematicians, as I’m sure there will be more breakthroughs in pure maths following.”