How DeepMind’s AI Helped Crack Two Mathematical Puzzles That Stumped Humans for Decades

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How DeepMind’s AI Helped Crack Two Mathematical Puzzles That Stumped Humans for Decades

Machine learning is aiming at the core of mathematics—a science for spotting patterns that lead to theorems about how our world works.

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With his telescope, Galileo gathered a vast trove of observations on celestial objects. With his mind, he found patterns in that universe of data, creating theories on motion and mechanics that paved the way for modern science.

Using AI, DeepMind just gave mathematicians a new telescope.

Working with two teams of mathematicians, DeepMind engineered an algorithm that can look across different mathematical fields and spot connections that previously escaped the human mind. The AI doesn’t do all the work—when fed sufficient data, it finds patterns. These patterns are then passed on to human mathematicians to guide their intuition and creativity towards new laws of nature.

“I was not expecting to have some of my preconceptions turned on their head,” said Dr. Marc Lackenby at the University of Oxford, one of the scientists collaborating with DeepMind, to Nature, where the study was published.

The AI comes just a few months after DeepMind’s previous triumph in solving a 50-year-old challenge in biology. This is different. For the first time, machine learning is aiming at the core of mathematics—a science for spotting patterns that eventually leads to formally-proven ideas, or theorems, about how our world works. It also emphasized collaboration between machine and man in bridging observations to working theorems.

“Human creativity enables mathematicians to instinctively understand where to look for emerging patterns,” said Dr. Christian Stump at the Ruhr University Bochum, who was not involved in the study but wrote an accompanying article.

Why Math?
I, like many others, still get panicky when thinking back to math classes in school. But what we learned there just scratched the surface of this fantastical world. Math isn’t just about numbers or algebra or geometry. It peeks into fundamental rules that may guide how our world works. Practically speaking, it laid the foundation that gave us computers and helped enable the AI algorithms that now power much of the online world.

The reason is that math tries to find patterns in data. Take one example: gravity. By examining how things fall—and on the shoulders of giants including Galileo—Isaac Newton took those observations, found patterns in them, and distilled those patterns into an equation. While that may sound boring, without that process we wouldn’t have flights, rockets, or space travel.

Math follows a cycle, said Stump. You start with a few relevant examples—say, the shape of things or dropping stuff from different heights—gather data, and then compute some of their properties and analyze the possible relationship of those properties until a pattern emerges. Mathematicians then keep testing these ideas in a more general or more complicated setting. If weird things pop up, then it’s time to update the pattern. The cycle continues and eventually leads to a new theorem.

This is great news in our digital world. We’re now producing data exponentially, meaning there’s been more data than ever to mine. The problem? It’s too much for any one mathematician to make sense of in his or her lifetime.

Enter AI
One thing AI is exceptionally good at is finding patterns in vast amounts of data.

Mathematicians have previously used software to help crunch numbers in their search for new theorems. But machine learning has been persona non grata, partly because it’s inherently probabilistic. Due to its design, these algorithms can only provide guesses, not certainty. And math requires certainty.

The solution? A man-machine tag team.

Reasoning that AI can provide insights that guide new mathematical ideas, DeepMind teamed up with Lackenby, Dr. András Juhász at Oxford University, and Dr. Geordie Williamson at the University of Sydney to probe two mathematical worlds: the theory of knots and the study of symmetries. Both address long-standing open questions that could influence our understanding of the world.

Take knot theory. On the surface, it’s about how a piece of rope ties into knots and what type of knots (important for both climbers and fishermen). But at its core, the theory contains mathematical principles that can help guide quantum computing—similar to how previous expansions from math and logic gave us our current computers.

Knot theory is especially alluring because different branches of mathematics—algebra, geometry, and quantum theory—share “unique perspectives,” wrote the DeepMind team in a blog post. But “a long-standing mystery is how these different branches relate.”

In the study, the team trained a machine learning model to bridge those connections. The AI was influenced by a trick in computer vision called saliency maps. Briefly, these maps are especially powerful at finding spots that carry more information—akin to the difference between a person’s eyes focusing in on something versus a random blurred-out backgrounds. Here, the maps pointed out especially interesting properties about geometry—a “signature”—that hint at an important aspect that’s previously been overlooked.

“Together [with Lackenby] we were then able to prove the exact nature of the relationship, establishing some of the first connections between these different branches of mathematics,” wrote the DeepMind authors.

In another proof of concept, DeepMind teamed up with Williamson to solve a problem in symmetries, which touches many other branches of math. Traditionally, mathematicians have studied it using charts or graphs. But like rendering a high-definition movie in 3D, the job quickly becomes too complicated, time-consuming, and even “beyond human comprehension.”

With a tailored AI, DeepMind discovered several interesting patterns in the field—so compelling that Williamson pursued them. He formulated a conjecture (something that’s apparently true based on all known data but remains to be proven with rigorous mathematics).

“I was just blown away by how powerful this stuff is,” said Williamson. “I think I spent basically a year in the darkness just feeling the computers knew something that I didn’t.”

What’s Next?
DeepMind has been steadily proving that machine learning isn’t just for games and play, but has a multitude of practical uses From solving core biological principles to predicting gene expression with AI, and now aiding mathematicians in their quest to find new theorems, AI is increasingly bolstering advancements in science.

But human intuition remains impossible to replicate. Due to the algorithm’s probabilistic nature—that is, it can only provide guesses—it’s on the human mathematicians to use existing methods to formally assess and prove the AI’s results. But the algorithm serves as a guide. Like a lighthouse, it points mathematicians in directions that are potentially correct. But ultimately, it’s up to the humans to use their judgment, intuition, and rigorous work to find the resulting breakthroughs. In this way, men and machines can propel each other’s learnings forward in a virtuous cycle.

For now, the AI has only been tested in limited cases. This particular AI can’t yet apply to all mathematical fields, partly because it’s relatively data-hungry. However, compared to many machine learning algorithms, it is energy efficient—enough to run on a laptop. And the mathematical community is “casually open-minded.”

“Neither result is necessarily out of reach for researchers in these areas, but both provide genuine insights that had not previously been found by specialists,” said Stump.

The DeepMind team is highly aware. “Even if certain kinds of patterns continue to elude modern ML, we hope our Nature paper can inspire other researchers to consider the potential for AI as a useful tool in pure maths,” they wrote. Their code is on Github for anyone to test.

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