A notorious maths problem first posed by Isaac Newton may be getting closer to a solution thanks to artificial intelligence. The three-body problem – the question of how three objects orbit one another under their own gravity – has baffled physicists and mathematicians for more than 300 years, but it turns out a neural network can find solutions remarkably quickly.
The three-body problem is difficult because it is a chaotic system, meaning it requires an extremely precise understanding of where the three objects start. For these systems, the “butterfly effect” comes into play – even a tiny error could result in an entirely different orbit. No single equation can yet predict how the objects will move and whether the orbits will be stable over time.
Instead, mathematicians have to meticulously test each scenario, either by hand or using computerised solvers, which can be slow and energy-intensive. Philip Breen at the University of Edinburgh, UK, and his colleagues have come up with a new way to solve it using a neural network, which can be up to 100 million times faster than the best computerised solvers.
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They trained their AI on a set of 9900 three-body scenarios generated by a state-of-the-art solver called Brutus. The researchers used 100 more scenarios from Brutus to make sure their system worked, and then 5000 unsolved scenarios to test it. It matched the examples from Brutus nearly exactly, demonstrating that the neural network could provide accurate and speedy solutions to the three-body problem.
The AI could improve our understanding of how black holes collide and form gravitational waves, says Breen.ĚýMany of those kind of complex dynamical systems can be boiled down to a series of three-body interactions that the neural network can easily solve, he says.
“It’s astonishing to me to find a totally new approach to this old problem,” says Douglas Heggie at the University of Edinburgh, who wasn’t involved with the research. One limitation is that the AI only works for a finite length of time, and if a particular three-body problem hasn’t been studied before, you don’t know in advance how long it will take to figure out what actually happens, he says.
The researchers have proposed a solution to this: rather than using the AI for the entire computation, just give it the hard bits, when the three bodies make close passes by one another. Then give the problem back to Brutus with that computational bottleneck already solved.
This could provide any number of solutions quickly, even without a neat equation, says Christopher Foley at the University of Cambridge, who also worked on the AI.
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“It’s less about the elegance and more about making progress and advancing our understanding of the building blocks of our physical environment,” he says. “If I can get the solutions, some would argue that it doesn’t matter how I get there as long as they are right.”
Even though the neural network can’t tell us how it gets its solutions, the ease with which it finds them might point us toward the three-body equation mathematicians have been seeking for centuries, Breen says.
“Because of how relatively small the neural network is with the amount of information it gives us, it actually kind of implies that there is perhaps an easier solution if one were to use higher dimensions. The network could be finding these tricks itself,” he says. “Just because something is chaotic, it’s not necessarily complex or unsolvable – maybe we haven’t actually found the right way to solve it.”
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