Two human mathematicians have made, with a āgreat deal of helpā from artificial intelligence, three important steps towards solving one of the worldās 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 ā whether that is used to design more efficient and stable aircraft wings, simulate blood flow through arteries or build space rockets ā despite their habit of occasionally breaking and outputting nonsense in certain scenarios.
Solving this problem ā working out whether the equations completely tally with the real world or whether their modelled smoothness and turbulence can deviate from it ā is one of the six remainingĀ 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 Harvard University and AI company Anthropic.
The pair claim three results: two of which were published along with Lean formalisation ā a process of converting a mathematical theory into computer code that allows it to be rigorously checked for logical flaws and errors ā and one that isnāt 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 arenāt yet generalised to the wider Navier-Stokes problem.
In the papers, they describe how they have taken previous work by Diego CĆ³rdoba and Luis MartĆnez-Zoroa to a conclusion with a āgreat deal of help from LLMsā, 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. āThis is a Deep Blue-Kasparov moment,ā he said, alluding to the famous chess match in 1996 where an IBM supercomputer beat chess world champion Garry Kasparov. Neither Buckmaster nor Alpƶge immediately responded to Ā鶹“«Ć½ās request for comment.
The results can be thought of as āstepping stonesā 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 āblow-upsā or turbulence can appear.
āItās a really hard problem because when it was stated, it wasnāt 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 itās not like youāre trying to prove something. You donāt know whether youāre trying to prove it or trying to find a counterexample,ā says Silvester.
Silvester says that expanding this result to Navier-Stokes wonāt 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.
āThere does not seem to be anything in principle preventing the methods from extending all the way to Navier-Stokes,ā wrote Tao. āAt 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.ā
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 ā 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 wonāt be enough and that there will be more to the solution than that, but she is optimistic. āIām confident, as Terence [Tao is], that this is a good way to go,ā 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.
āNothing will change in the applications where [Navier-Stokes] is used because of this result,ā says Silvester. āItās a mathematical nicety, honestly.ā
