Scientists question OpenAI's method in solving Navier-Stokes mathematical problem

OpenAI claimed that its artificial intelligence (AI) agents managed to solve one of the most relevant and still unanswered mathematical problems: an issue related to the Navier-Stokes equations, which model fluid movement. This potential solution could secure a $1 million prize (equivalent to about R$ 5.4 million), but mathematicians are raising doubts about the speed of the progress and the methodology employed by the company.

The controversy intensified after Tristan Buckmaster, a mathematician from New York University (USA), stated that he was questioned by OpenAI collaborators about his work on the same problem. In a meeting held on September 6, he reported receiving the inquiry: 'Why would you ruin your career?'

Buckmaster detailed this incident in a public statement, expressing suspicions that OpenAI might have taken advantage of research developed by him and Levent Alpöge, a mathematician from Anthropic.

The Navier-Stokes equations were conceived in the early 19th century by Claude-Louis Navier and George Gabriel Stokes. They are fundamental for describing fluid dynamics, helping to explain phenomena ranging from the airflow generated by propellers to blood circulation in arteries.

The core of the mathematical challenge lies in determining whether these equations will always produce finite results or if, under certain circumstances, they can exhibit a singularity, known as 'blowup'. In this scenario, a physical variable, such as pressure or fluid velocity, could grow infinitely at a specific point in space and time.

This question is part of the Millennium Prize Problems established by the Clay Mathematics Institute, where each correct solution offers a $1 million prize (approximately R$ 5.4 million). To investigate the topic, OpenAI applied a technique called forcing, which consists of introducing a smooth and controlled perturbation into the simulated fluid.

The company mobilized about ten thousand AI agents to work simultaneously on the problem, keeping them active for approximately 88 hours. Initially, about 100 agents dedicated themselves to analyzing the Euler equations, a variation of the Navier-Stokes equations that ignores viscosity.

After about 50 hours of processing, the agents identified instances of blowups in the Euler equations. This discovery motivated the team to focus their efforts on the Navier-Stokes problem. OpenAI claims that its agents generated a singularity where a vortex becomes progressively narrower, elongated, and thin, causing the fluid to rotate towards infinity, mathematically speaking.

The proof was sent to Lean, a system used to validate every step of the mathematical arguments. According to OpenAI, the agents exchanged 2.7 million messages and generated about 130 billion tokens.

In a press conference held on Tuesday (8), company representatives informed that a similar task executed for a client would cost $15 million (about R$ 81 million).

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Sam Altman, CEO of OpenAI, admitted that the company became aware of the work of Buckmaster and Alpöge before disclosing its own finding. He commented: 'We were curious to see if ours could also succeed.' Altman added that the methodologies 'seem different,' which Buckmaster contests.

OpenAI mentioned the two mathematicians in its communication, recognizing the independent resolution of the Euler problem using forcing, and offered them the opportunity to publish the work jointly.

The controversy also involves the use of Codex, OpenAI's programming assistant. Buckmaster stated that he used this tool during his research, inputting drafts of the work done with Alpöge into it. The mathematician declared that he asked OpenAI if the Codex sessions could be accessed or used to train models, but received no answer on this matter.

OpenAI guaranteed that employees or agents of the company had not accessed the couple's work 'by any means' before its public disclosure. However, the company acknowledged that it could not rule out the possibility that anonymized data from the use of its products contributed to the improvement of its models.

Buckmaster also alleged that Sébastien Bubeck, responsible for mathematical research at OpenAI, pressured twice to remove Alpöge from the authorship of a paper the two mathematicians were preparing. Alpöge belongs to Anthropic, a competitor of OpenAI.

Bubeck refuted the accusations, classifying the allegations as 'false and inflammatory.' He maintained that OpenAI did not use the researchers' prompts or proofs to guide its models.

For his part, Buckmaster stated that he had not seen the proof presented by OpenAI and does not know how the model reached the result or if the researchers' data was used. He requested an independent investigation into the origin of the company's work.

Diego Córdoba expressed surprise at the speed of the result's emergence. Other mathematicians also voiced apprehension about the way the discovery was communicated.

Terence Tao, a mathematics professor at the University of California, Los Angeles (UCLA), observed that the episode demonstrates how the mere rumor that someone is working on a major problem can drive a massive AI effort to solve it before the original research is finalized. According to Tao, financial incentives can lead researchers to withhold promising ideas from the scientific community, harming a secular tradition of open science.

OpenAI's proof is still awaiting formal peer review. The Clay Mathematics Institute has neither confirmed nor officially rejected the solution presented by the company. The article was initially published in Olhar Digital.

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olhardigital.com.brSeptember 8, 23:13

OpenAI claims to have solved the Navier-Stokes problem, a Millennium Prize challenge

OpenAI announced that it has found a solution to one of the most complex and ancient problems in mathematics: the existence and smoothness problem of Navier-Stokes, which is part of the seven Millennium Prize Problems established by the Clay Mathematics Institute.

According to the company, an internal artificial intelligence (AI) system generated an analytical proof demonstrating that the Navier-Stokes equations can exhibit a singularity at a specific time. OpenAI also made a formalization of this proof available in the Lean language, which is used for mathematical verification of the result.

This announcement marks a notable progress in the application of AI to mathematics. The Navier-Stokes equations are crucial for modeling fluid motion, being applied in fields such as weather forecasting, aircraft design, and blood flow study.

The question that remained unanswered for decades was whether these equations maintained a smooth solution in three dimensions or if, under certain conditions, they could collapse and generate a singularity, a moment when fluid velocity would increase infinitely in a finite period.

The problem was included in the seven Millennium Prize Problems in 2000, with a prize of US$1 million (equivalent to R$5 million) for anyone who presented a correct solution.

How the solution was achieved

To reach this result, OpenAI employed a system composed of coordinated AI agents. It is estimated that about ten thousand agents worked simultaneously on solving Navier-Stokes, receiving variations of the problem and being distributed into groups to explore different methods.

The process began after OpenAI researchers heard rumors on September 1st that two millennium problems might have been solved. The company then decided to test its new internal model on the remaining challenges.

The system was also used to investigate a similar question related to Euler's equations, which do not contain the viscosity term present in Navier-Stokes. About 100 agents dedicated approximately 50 hours to solving this second problem.

After obtaining this initial result, OpenAI directed its efforts toward Navier-Stokes. The agent groups began exchanging findings, using Codex to consolidate the most relevant information among the various teams.

The agents finally reached the Navier-Stokes solution on September 5th, about 88 hours after the start of the work. The formalization and validation phase of the proof in Lean consumed another 17 hours.

During all tests conducted in the project, the agents sent 4.9 million messages and consumed approximately 300 billion tokens. Specifically in the Navier-Stokes work, 2.7 million messages and about 130 billion tokens were recorded.

The computational cost was also substantial. According to supporting material, the operation demanded processing power estimated in millions of dollars due to the vast amount of resources required to operate so many AI systems simultaneously.

Although it considers the finding as a resolution of the Navier-Stokes problem, OpenAI stated that it does not intend to claim the US$1 million prize offered by the Clay Mathematics Institute.

The company released the work as a demonstration of the advancement of its AI models, publishing both the description of the solution and its formalization in Lean, allowing other mathematicians to examine the result.

Scientific Dispute Involved

However, the announcement did not end the debate about the achievement. In addition to the mathematical analysis of the proof, a controversy arose involving researchers working on related topics and the possibility that ideas developed by them had reached OpenAI.

OpenAI's announcement occurred amidst a scientific dispute involving researchers linked to the company itself and Anthropic. Tristan Buckmaster, a mathematics professor at New York University (USA), and Levent Alpöge, a researcher at Anthropic, were working on issues related to fluid dynamics.

Both researchers had presented results obtained with 'significant help' from language models, including systems from Anthropic and OpenAI. Their work focused on Euler's equations and presented a phenomenon called 'blow-up', or singularity. Part of these results had also been formalized in Lean.

The situation took on the contours of a dispute over credit when Buckmaster alleged that rumors about the method developed by his team might have been transmitted to OpenAI. He maintained that an internal team at the company had used its own model to advance in the complete Navier-Stokes problem.

There was also discussion about authorship. Buckmaster stated that OpenAI offered him exclusive authorship on a paper acknowledging that the internal model had solved the problem, but without including Alpöge, who works for a competing company.

OpenAI denied having accessed the researchers' work before its public disclosure. The company assured that its researchers and agents did not view the material from Buckmaster and Alpöge by any means before publication.

In its statement, OpenAI added that, upon contacting the two researchers after completing its own work, it discovered that they had solved a distinct version of the problem: the case of Euler's equations with external force. The company recognized the researchers' priority in this result.

This advancement also reignites the debate about the role of AI in generating scientific knowledge. AI systems are being trained to solve mathematical problems through techniques such as reinforcement learning. By repeatedly interacting with problems whose answers can be objectively verified, the models learn which strategies are effective and which lead to errors.

The Lean language enhances this process by enabling mathematical proofs to be converted into code and submitted to formal verification.

OpenAI's result is particularly relevant because the Navier-Stokes problem was among the most difficult in modern mathematics. For decades, mathematicians sought to determine whether the equations could develop a singularity or remain always smooth.

Thus, the feat transcends a mere demonstration of computational capability; it brings AI closer to a function that was long considered exclusively human: discovering new approaches to fundamental mathematical questions.

Simultaneously, the dispute between OpenAI and researchers linked to Anthropic raises a question that may become increasingly crucial as AI actively participates in scientific research: who holds the authorship of a discovery when humans and machines collaborate—and how to protect unpublished ideas while they are being developed with the aid of large AI laboratories?