Tuesday, September 8 will go down as a milestone in artificial intelligence alongside IBM’s Deep Blue defeating world chess champion Gary Kasparov in 1997 and AlexNet winning the 2012 ImageNet competition. For the first time, an AI model was used to solve one of math’s most famous problems — a Millennium Problem. OpenAI announced that a swarm of 10,000 agents had produced a solution to a problem known as Navier-Stokes.
Unlike those earlier AI milestones, however, no one was happy on Tuesday. Not even OpenAI.
The night before OpenAI’s announcement, the NYU mathematician Tristan Buckmaster announced that he and collaborator Levent Alpöge had solved three problems closely related to Navier-Stokes. Alongside rough drafts of three papers — some 245 pages in total — Buckmaster released a statement excoriating OpenAI.
“I had planned to say on announcing our work that the results are not the important thing. Rather the important thing is instead the significance that a mathematician and an LLM model can now do all this work in a month,” Buckmaster wrote on Monday. But “instead of these incredibly important developments, I find myself writing about something else.”
Buckmaster wrote that he initially reached out to a mathematician at OpenAI on Thursday, September 3. Rumors were swirling that Anthropic had solved two Millennium Problems. Buckmaster wanted OpenAI to know that those rumors likely referred to his effort, which wasn’t officially supported by Anthropic. His collaborator Alpöge was an Anthropic employee, Buckmaster said, but was working on the project in his spare time.
Three days later, Buckmaster spoke with that mathematician and Sébastien Bubeck, who led the OpenAI effort to solve Millennium Problems. OpenAI’s approach to Navier-Stokes used the same broad approach as Alpöge and Buckmaster’s, which Buckmaster wrote “almost nobody” was working on. OpenAI began work after the rumours about an Anthropic effort reached OpenAI.
Through a massive computational effort, OpenAI had beaten Buckmaster and Alpöge to a full Navier-Stokes solution. According to Buckmaster, Bubeck offered to merge their efforts and let Buckmaster write a paper announcing the full Navier-Stokes result, as long as the paper acknowledged that an OpenAI model had solved it. But under this offer, Alpöge, who works for Anthropic, would not be a co-author.
Buckmaster was furious and went public with the story.
Predictably, the drama overshadowed the mathematics. People debated whether Buckmaster was right to be outraged — or whether Bubeck’s response exonerated OpenAI.
But I think that focusing on the details of the drama risks missing the larger point.
Mathematics is as much about cultivating a community of experts as it is about solving individual problems. Some of OpenAI’s behavior might have been reasonable in the context of competing with another well-resourced company like Anthropic. But spending millions of dollars on compute and thereby scooping an academic researcher breaks the norms of the academic math community.
If everyone behaved like OpenAI, mathematicians would have to keep their work secret until it was ready for publication. And mathematicians value openness and collaboration. So it’s considered bad form for someone to learn that another mathematician has had some promising early results and then sprint to complete the work first.
OpenAI sought to gain prestige by solving a famous math problem. But the way it went about that has arguably undermined the community that made solving the problem prestigious in the first place.
The Navier-Stokes problem is important but useless
The Navier-Stokes equations are a way of describing how a fluid — such as water in a stream or air in the atmosphere — moves through space.
Fluid dynamics is very complicated, and in most cases there’s no explicit formula that can calculate where the fluid will be at every point in the future. Instead, Navier-Stokes equations describe how the fluid’s motion is changing by calculating the direction and speed at which each point in the fluid is moving at a specific point in time.
In the illustration below, every point has an arrow describing its direction — the direction and length of these arrows are described by Navier-Stokes equations.
To model a fluid’s movement over time, scientists can move forward through time in small steps. At each step, they use the Navier-Stokes equations to estimate how the velocity field changes, move the fluid forward slightly, and repeat. The result looks something like this:
This is a useful way to simulate the movement of a liquid. But it gives rise to an interesting theoretical question: are there situations where fluid movement predicted by the Navier-Stokes equations leads to absurd outcomes?
OpenAI found a fluid arrangement in 3D where the Navier-Stokes equations lead to absurd outcomes, but the company’s solution is hard to draw. So to help give readers an intuition for what this means, here’s an example of where a simpler model of fluid motion breaks down:
At first, we have a normal water wave. But as the simulation progresses, the leading edge gets so steep that water particles basically have to teleport in order to reach their positions in time. A real wave wouldn’t behave like this.
OpenAI found a situation where the Navier-Stokes equations give rise to another type of mathematical breakdown — called a singularity — in a scenario where the liquid is acted on by a specially chosen smooth force.1 As the company wrote in the announcement: “The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics.”
Eventually, in the central region, “the velocity of the fluid grows without bound.” Again, this would never happen in a real fluid.
Alpöge and Buckmaster found a similarly implausible outcome for three related and somewhat simpler models of how liquids flow.2
While this is a significant breakthrough for mathematicians, it probably won’t have much practical significance. On September 3, the mathematician Terence Tao wrote a Mastodon thread about the implications of an AI model solving Navier-Stokes. He wrote:
While the equations do come from a very natural physical motivation - the study of incompressible fluids - the regularity problem is not important for its direct physical application. Computational fluid dynamics is already a mature subject, deployed extensively in the atmospheric sciences, for instance, and its empirical capabilities and limitations are already well understood. A theoretical guarantee of regularity, or conversely a pathological instance of blowup, for these equations would be intellectually interesting for such applications, but would not radically transform the way we would, for instance, model weather prediction or climate change.
The process of finding the solution
If it’s not useful for physical applications, why should anyone care?