Yesterday, OpenAI announced that an internal model had successfully created a proof that showed Navier-Stokes equations could, under certain conditions, “blow up” and produce results that were not physically possible.
This problem has vexed mathematicians for over 90 years. It was, and currently still is, one of the Millennium Prize Problems. Those are a series of seven mathematical problems that the Clay Mathematics Institute has set up $1 million prizes for anyone who can solve them.
These problems, first established in May 2000, are all problems that have been unsolved for decades or centuries and are considered to be at the frontier of mathematics. If this solution is accepted, it would be only the second time that a Millennium Prize Problem has been solved.
According to OpenAI, the solution came after an 88-hour run that spent more than 300 billion output tokens. In total, they spent roughly $22.5 million in compute across their entire attempt.
This moment should have been a major coup for OpenAI. However, it quickly became mired in controversy. That’s because two human researchers have raised concerns about and raised questions about who really deserves the credit for the breakthrough.
Though everyone has stopped short of saying that this is a work of plagiarism, its very clear that there are serious concerns about authorship and credit. To make matters worse, these issues are only going to get worse as the use of AI becomes more widespread.
The Story So Far
The Navier-Stokes equations are a set of mathematical equations that describe the motion of fluids, including gases. They are widely used in various engineering and scientific fields, such as determining the flow of air around an airplane wing or the flow of blood through a human body.
Though the equations have proven reliable, it is unknown if the equations “blow up” or not. An equation blows up when it produces a result that is not physically possible, usually an infinite value. It speaks to the heart of the Navier-Stokes equations as they may not always be completely accurate.
The Euler equations are a simplified version of the Navier-Stokes equations. They also describe the motion of fluids, but they do not account for the effects of viscosity. They are older than the Navier-Stokes equations, but are seen as a stepping stone to solving the Navier-Stokes equations.
NYU professor Tristan Buckmaster and Anthropic employee Levent Alpöge, who was working in his personal capacity, were working on the Euler equations and, according to Buckmaster, had solved them. They had used AI heavily in their work, including the use of Anthropic’s Claude and OpenAI’s Codex tool and Astra model.
According to OpenAI, on September 1, they heard a rumor that the Navier-Stokes equations had been solved. They decided to test their unnamed internal model by throwing it at the problem. They claim that, after 50 hours, they had produced a proof for the Euler equations. They then turned to the Navier-Stokes equations and spent another 88 hours producing a proof of those equations.
Buckmaster claims that, on September 3, he heard a rumor that Anthropic had solved a major math problem. He said he reached out to OpenAI to clarify that the work was a personal collaboration between him and Alpöge, not institutional.
Then, on September 6, one day after OpenAI finished its proof but was waiting on verification, Buckmaster (without Alpöge) discussed OpenAI’s results with OpenAI’s Sebastien Bubeck. Here things divide.
Buckmaster claims that Bubeck offered him two options. The first was to be listed as the sole author of the paper, crediting OpenAI’s models (and omitting Alpöge), or do a joint release with OpenAI, also removing Alpöge. Buckmaster refused both options. He goes on to say that Bubeck asked him, “Why would you ruin your career?” and that, “If you don’t want me to be nice, then I don’t have to be nice.”
Bubeck (and OpenAI more broadly) say that they did not realize that Buckmaster and Alpöge had only solved the Euler equations. Bubeck strongly asserted that OpenAI had not seen their work previously, but could not say for certain if their model had been trained on Buckmaster and Alpöge’s work since it had been uploaded to the service. Buckmaster also raised questions about how much OpenAI’s work relied on human involvement, both at OpenAI and in previous mathematicians’ work.
Both sides published dueling statements on the matter. However, as of this writing, very little is resolved. Though both sides have published their proofs (or at least some of them), the analysis of them will take a long time.
This is one of those times the truth will likely only come out in hindsight.
The Big Concerns
To be clear, this is not a story about AI vs. humans or AI vs. no-AI. Both sides made heavy use of AI in their work. Instead, there are two core questions here.
First, did OpenAI train its model on Buckmaster and Alpöge’s work and, if so, what role did it play in their proof?
OpenAI points out that they solved the Euler equations using a completely different method than Buckmaster and Alpöge. However, Buckmaster points out that OpenAI’s resolution to the Navier-Stokes equations used the same approach as them. They said that the approach was not obvious and was actually based on previous work by Diego Córdoba and Luis Martínez-Zoroa.
While Buckmaster and Alpöge’s work did use AI, it followed the more traditional academic process. AI was used to supplement their research, not replace it. They based their work on previous research and worked, over years, to tackle the problem. OpenAI, on the other hand, spent a week and millions of dollars in compute to solve the problem. But, here’s the problem. Even if OpenAI didn’t train on Buckmaster and Alpöge’s work, it was still trained on countless other mathematicians’ work. The OpenAI paper cites 29 sources, but we have no way of knowing if that source list is close to complete. In fact, OpenAI has no way of knowing if it’s complete. (Note: Córdoba and Martínez-Zoroa are cited in the OpenAI paper.)
This isn’t a purely academic issue. While credit is important, in academia, it’s also part of the process that removes faulty work. If something OpenAI relied upon turns out to be wrong or otherwise faulty, it could jeopardize their conclusions. However, we can’t know if that happens without full citations.
The second problem is more open ended and theoretical. But, if we assume that OpenAI did train on Buckmaster and Alpöge’s work, then we have to ask a simple question: What’s stopping OpenAI, or any other AI company, from using their massive compute power to beat human academics to their discoveries?
Since Buckmaster and Alpöge were not doing institutional work, their efforts were hamstrung by a lack of resources. According to Buckmaster, he paid for much of the compute time he used himself. OpenAI, on the other hand, simply dropped millions of dollars in compute time on the problem based on a rumor alone.
When computing time becomes a limiting factor in the research, those who have the most access to those resources can simply outpace others. It’s easy to imagine a scenario where Buckmaster and Alpöge’s had successfully solved, or were close to solving, the Navier-Stokes equations. OpenAI, hearing this rumor, threw their resources at the problem and beat the duo to publication.
This is grossly unfair because OpenAI’s work is based on the work of countless uncredited humans and because it’s using money and resources as a substitute for skill. Imagine, for a moment, if Buckmaster and Alpöge had been able to spend that much money on compute? These are simply not resources that colleges, let alone individuals, have access to.
This positions AI companies not just as gatekeepers of research, but as a potential threat to human academics who are putting in the time and effort to solve difficult problems.
Bottom Line
From a technical perspective, there’s no doubt that what OpenAI did was amazing. Even as an AI skeptic, a tool that can solve a problem as complex as the Navier-Stokes equations in just over a week is impressive. In his paper, Buckmaster refers to this as a “Deep Blue-Kasparov” moment. 1This is a reference to the 1997 chess match between the IBM computer Deep Blue and the reigning world chess champion Garry Kasparov. Deep Blue beat Kasparov over the course of the six games and it was widely seen as the moment computers surpassed humans in chess.
While I don’t think that the analogy is perfect, it’s easy to see why he says it. If it’s accepted as a solution, OpenAI was able to quickly solve a problem that had vexed mathematicians for over 90 years. But everything that OpenAI’s model knows about mathematics is based on the work of humans. Without the work of those humans, OpenAI’s model would not be able to solve even a basic equation.
However, this introduces a new problem. If AI makes a discovery or a breakthrough, who gets the credit for it? The person who typed the prompt? The person who trained the model? The person whose work was used most heavily by the model? There’s no easy answer.
But it’s an important one to solve. Not just because credit is important, but because the nature of math and science requires being able to trace the knowledge that a new discovery is built on.
As impressive as what OpenAI did is, it may literally create more serious problems than it solves.
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