Terence Tao says automated proof checking is why the AI labs stopped consulting mathematicians

OpenAI announced this week that its AI agents found that the Navier-Stokes equations do sometimes blow up, the phrase describing where part of a fluid starts to move infinitely fast.

Steven Brunton at the University of Washington says: "Every equation is an approximation."

The implication of the result is not that fluids will start misbehaving but that the equations are not perfect.

"Part of the reason why these AI companies have become so aggressive and are outputting so much is because they have formalisation to check their output," says Terence Tao at the University of California, Los Angeles.

Tao says that as recently as a year or two ago AI companies had genuine dialogue with mathematicians to help understand and check their AI-generated work, but with advances in AI formalisation they are now pressing ahead alone and "getting all these increasingly meaningless accomplishments".

Tao says formalisation is a double-edged sword, with obvious benefits but also big downsides, such as the rapid way results are being dropped on academia with no follow-up or engagement, which he has criticised as harming the field.

AIs tend to produce hard-to-follow and convoluted arguments; for the sake of only proving a theorem this does not really matter because of formalisation, but for the sake of gaining insight it is lacking.

To find the answer, OpenAI used a technique called forcing, giving the simulated fluid a smooth, controlled nudge from the outside.

The specific blow-up was a spinning whirlpool that becomes longer, thinner and more intense over time before its swirling becomes infinite, and the new proof showed that this happens in finite time and using only a finite amount of force on the fluid.

This week Tristan Buckmaster at New York University and Levent Alpoge at Anthropic published a breakthrough removing that force requirement for the Euler equations and several related situations, and a few hours afterwards OpenAI published its full result.

Buckmaster said that on Aug. 15 he and Alpoge achieved blowup results for the Euler equations with smooth forcing, and that on Sept. 3 he received word through Alpoge that OpenAI had caught wind of their progress.

John Evans at the University of Colorado Boulder says of the OpenAI proof: "Even though it's 166 pages of complicated math, you can actually draw it down to a couple of simpler physical mechanisms."

Evans says AI agents could help make better models of fluid flow in the future.

In 2014 Thomas Hou at the California Institute of Technology and colleagues showed one way to create a blow-up for the Euler equations, but their approach only applied when the fluid was walled in.

In 2021 Luis Martinez-Zoroa at CUNEF University and Diego Cordoba developed a method for creating blow-ups that worked for both the Euler and Navier-Stokes equations, dividing the fluid into thin layers and adding the layers up to produce a cascade of effects.

Martinez-Zoroa and Cordoba's work relied on an unrealistic force being used, which means it would not solve the Millennium problem.

"AI [use] was absolutely pivotal and the scale [of it] is interesting and useful, but the solution approach was developed by brilliant humans," says Steven Brunton at the University of Washington.

Evans says the AI's trick was ultimately distinct, not just a more powerful repetition of the Martinez-Zoroa and Cordoba approach.

OpenAI has already said it has made "substantial progress" on another of the Millennium Prize problems.

"If the world's moving so fast, there won't be anything that they can do slowly because it'll all get done before they finish," says Helen Wilson at University College London, about PhD training.

"If there are no mathematicians then those results don't go anywhere. It's just an industry of machines producing papers that nobody reads," says Wilson.

Wilson says: "I'm in the 'a bit terrified' camp."

PhD students are currently often given a relatively low-level problem to chew over for a few years and hone their skills on.

The $15 million customer price divided by the 88-hour run is about $170,000 per hour of wall clock.

The $15 million customer price divided by approximately 130 billion output tokens is about $115 per million output tokens.

The Navier-Stokes blow-up question appears on the list of Millennium Prize Problems posed by the Clay Mathematics Institute, and a solution comes with a $1 million reward.

OpenAI set roughly 10,000 AI agents to work simultaneously on the problem for about 88 hours, with 100 of them starting on the related Euler equations.

Buckmaster claimed that during a Sept. 6 meeting with OpenAI employees he was issued a stark warning: "Why would you ruin your career?", an exchange he described in a detailed public statement this week.

After roughly 50 hours the agents found blowups in the Euler equations, which convinced OpenAI that Navier-Stokes was within reach, and they turned to the bigger problem.

OpenAI began working on the problem after it got wind that a pair of mathematicians had made some progress that had not yet been published.