Major breakthrough made on famous Millennium maths problem | New Scientist

Two human mathematicians have made, with a “great deal of help” from artificial intelligence, three important steps towards solving one of the world’s most famous outstanding mathematical problems. Other researchers say the approach could lead to a full solution in time and that it may already be enough to land the pair a $1m Millennium Prize.

The Navier-Stokes equations have been used to model the flow of fluids for two centuries – whether that is used to design more efficient and stable aircraft wings, simulate blood flow through arteries or build space rockets – despite their habit of occasionally breaking and outputting nonsense in certain scenarios.

Solving this problem – working out whether the equations completely tally with the real world or whether their modelled smoothness and turbulence can deviate from it – is one of the six remaining Millennium Problems, the thorny mathematical puzzles published by the Clay Mathematics Institute that will net the solver a $1 million prize.

Now, Tristan Buckmaster at New York University has announced groundbreaking results that edge us towards that larger solution via a document uploaded to his website, developed with Levent Alpöge at AI company Anthropic.

The pair claim three results: two of which were published along with Lean formalisation – a process of converting a mathematical theory into computer code that allows it to be rigorously checked for logical flaws and errors – and one that isn’t yet published as the pair await a finished formalisation. The findings relate to close cousins of Navier-Stokes, the Boussinesq approximation and the Euler equations, but aren’t yet generalised to the wider Navier-Stokes problem.

In the papers, they describe how they have taken previous work by Diego Córdoba and Luis Martínez-Zoroa to a conclusion with a “great deal of help from LLMs”, including large language models from Anthropic and OpenAI.

In his announcement, Buckmaster said he sees the results as being less important than the fact that AI is rapidly becoming a powerful amplifier of human mathematical effort and leading to faster progress. “This is a Deep Blue-Kasparov moment,” he said, alluding to the famous chess match in 1996 where an IBM supercomputer beat chess world champion Garry Kasparov. Neither Buckmaster nor Alpöge immediately responded to New Scientist‘s request for comment.

The results can be thought of as “stepping stones” to a full solution of Navier-Stokes, says David Silvester at the University of Manchester, UK.

Silvester says the paper focusing on the Euler equations shows that spontaneous “blow-ups” or turbulence can appear.

“It’s a really hard problem because when it was stated, it wasn’t clear whether the result was true: that is that there are smooth solutions and it stays forever stable, or, in fact, there is some blow-up. So it’s not like you’re trying to prove something. You don’t know whether you’re trying to prove it or trying to find a counterexample,” says Silvester.

Silvester says that expanding this result to Navier-Stokes won’t be trivial, but that the current work alone may be enough for the pair to claim a Millennium prize.

Terence Tao at the University of California, Los Angeles, wrote in a social media post that he believes the work takes us very close to a full Navier-Stokes solution.

“There does not seem to be anything in principle preventing the methods from extending all the way to Navier-Stokes,” wrote Tao. “At this point, I would not be surprised if one could batter out such an extension by pouring an enormous amount of compute and AI assistance at such a task.”

Camilla Nobili at the University of Surrey, UK, says that the key question now is whether a similar example can be found in Navier-Stokes, which builds upon Euler equations by adding more detail like friction and dissipation – elements that have a natural tendency to smooth out simulations and keep them more regular. She believes that running the same technique on Navier-Stokes won’t be enough and that there will be more to the solution than that, but she is optimistic. “I’m confident, as Terence [Tao is], that this is a good way to go,” says Nobili.

Despite the fame and long-standing insolubility of Navier-Stokes, a working result is unlikely to deliver any practical benefit, says Silvester. Computer models on fluid dynamics are already so good that they have rendered wind tunnels largely obsolete.

“Nothing will change in the applications where [Navier-Stokes] is used because of this result,” says Silvester. “It’s a mathematical nicety, honestly.”