An artificial intelligence tool that hunts for unintuitive mathematical patterns appears able to tackle a wide range of maths problems, including some that haven’t seen progress for decades.
AI models have typically struggled at solving the sorts of maths problems that stump humans, although there have been some successes on specific problems. Because of this, most mathematicians have been reluctant to devote the extra time to learning to use AI as part of their research.
Now, at the Fundamental AI Research (FAIR) team at Meta – the parent company of Facebook – and his colleagues have come up with an AI tool, called PatternBoost, that appears able to solve many more problems and requires less effort to set up. “This is a general idea,” says Charton. “The whole surprise is that it actually works. You can apply it to a fairly large set of problems.”
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PatternBoost works in two phases. The first phase, called local search, uses an algorithm seeded with randomness to generate possible solutions to a problem, identifying those that seem most promising. These are then passed to an AI model, which uses the same transformer algorithm that powers ChatGPT to study them and produce more in the same vein. These results are then fed back into the local search algorithm, and the process is repeated until the best possible solution is found.
All of the problems that Charton and his team tried PatternBoost on were from a mathematical field called extremal combinatorics, which looks to find very large or small mathematical objects that obey some sort of rule. One was called the no-sphere problem, which involves finding the largest collection of points that fit on a three-dimensional grid, while ensuring that there is no set of five points that sit on a sphere within that grid.
“Everybody was very, very certain that the solution was 17, so that you couldn’t have more than 17 points,” says Charton. “I was just trying out experiments [with PatternBoost] and I found an 18 — the mathematicians didn’t really believe it. It’s not an extremely researched problem, so maybe it’s not that spectacular, but at least it surprised the specialists.”
The researchers tried problems from combinatorics because they were easy to fit into PatternBoost’s structure, but it is possible that it could be applicable to many different mathematical problems, says team member at the University of Wisconsin-Madison. “We have extremely little intuition for what kinds of problems are going to be effectively approached by this method. Maybe it’s all problems, who knows.”
It is an innovative idea, says at Imperial College London. “The most exciting thing about it is this has clearly got more potential,” he says. “They’ve given it certain questions in combinatorics — it’s not going to revolutionise all math — but there are certain areas of math where you can absolutely imagine this idea gives you an extra edge.”
arXiv
