One of the trickiest problems in mathematics has fallen to the hands of an advanced AI model. Following the announcement that OpenAI has resolved one tantalising aspect of the Navier-Stokes equations, researchers are now urgently trying to understand its implications. The Navier-Stokes equations were first written down around 200 years ago and describe the behaviour of fluids.
They’ve been instrumental in everything from building better airplane wings to creating effective artificial hearts. Yet, our understanding of the equations is so limited that resolving a relatively simple question about them is one of the prestigious Millennium Prize Problems, where a solution means a $1 million reward. That question is if the Navier-Stokes equations always work or if they sometimes break down and no longer make sense.
Terence Tao: AI companies are harming mathematics The team at OpenAI has found that the equations do sometimes fall apart, finding situations known as “blow-ups”, where the speed of part of the fluid suddenly becomes infinity. It would be like an infinite whirlpool suddenly appearing in your bath. The OpenAI team found the answer thanks to more than 10,000 AI agents working together in different configurations.
About 88 hours later, they had a solution. In 2014, Thomas Hou at the California Institute of Technology and his colleagues showed one way to create a blow up for Euler equations, one of the steppingstones towards Navier-Stokes. However, their approach only applied when the fluid in question was walled in and not more generally.
Other teams tried to extend it but without success. Researchers started to suspect that the Navier-Stokes equations really can blow up, but no-one had definitively shown how to construct a specific fluid flow that does so. What’s next for mathematics now that AI is upending the field?
A breakthrough came in 2021 when Luis Martínez-Zoroa at CUNEF University in Spain and Diego Córdoba at the Spanish National Research Council developed a new method for creating blow ups that worked for both Euler and Navier-Stokes equations. Their key insight was to divide the fluid into thin layers and perform calculations for each. Adding up the layers resulted in a cascade of effects that can ultimately cause a blow-up.
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