Two human mathematicians have made, with a 鈥済reat deal of help鈥� from artificial intelligence, three important steps towards solving one of the world鈥檚 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 鈥� whether that is used to design more efficient and stable aircraft wings, simulate blood flow through arteries or build space rockets 鈥� despite their habit of occasionally breaking and outputting nonsense in certain scenarios.
Solving this problem 鈥� working out whether the equations completely tally with the real world or whether their modelled smoothness and turbulence can deviate from it 鈥� is one of the six remaining聽Millennium Problems, the thorny mathematical puzzles published by the Clay Mathematics Institute that will net the solver a $1 million prize.
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Now, at New York University has announced groundbreaking results that edge us towards that larger solution, developed with at AI company Anthropic.
The pair claim three results: two of which were published along with Lean formalisation 鈥� a process of converting a mathematical theory into computer code that allows it to be rigorously checked for logical flaws and errors 鈥� and one that isn鈥檛 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鈥檛 yet generalised to the wider Navier-Stokes problem.
In the papers, they describe how they have taken previous work by Diego C贸rdoba and Luis Mart铆nez-Zoroa to a conclusion with a 鈥済reat deal of help from LLMs鈥�, 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. 鈥淭his is a Deep Blue-Kasparov moment,鈥� he said, alluding to the famous chess match in 1996 where an IBM supercomputer beat chess world champion Garry Kasparov. Neither Buckmaster nor Alp枚ge immediately responded to New 女生小视频鈥榮 request for comment.
The results can be thought of as 鈥渟tepping stones鈥� to a full solution of Navier-Stokes, says at the University of Manchester, UK.
Silvester says the paper focusing on the Euler equations shows that spontaneous 鈥渂low-ups鈥� or turbulence can appear.
鈥淚t鈥檚 a really hard problem because when it was stated, it wasn鈥檛 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鈥檚 not like you鈥檙e trying to prove something. You don鈥檛 know whether you鈥檙e trying to prove it or trying to find a counterexample,鈥� says Silvester.
Silvester says that expanding this result to Navier-Stokes won鈥檛 be trivial, but that the current work alone may be enough for the pair to claim a Millennium prize.
at the University of California, Los Angeles, that he believes the work takes us very close to a full Navier-Stokes solution.
鈥淭here does not seem to be anything in principle preventing the methods from extending all the way to Navier-Stokes,鈥� wrote Tao. 鈥淎t 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.鈥�
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 鈥� 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鈥檛 be enough and that there will be more to the solution than that, but she is optimistic. 鈥淚鈥檓 confident, as Terence [Tao is], that this is a good way to go,鈥� 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.
鈥淣othing will change in the applications where [Navier-Stokes] is used because of this result,鈥� says Silvester. 鈥淚t鈥檚 a mathematical nicety, honestly.鈥�
