OpenAI announced that one of its artificial intelligence models had solved two of the four questions related to Navier–Stokes equations whose solutions carry a one-million-dollar prize. Shortly beforehand, a group of mathematicians had made significant progress toward solving the same problem. So, who deserves the credit?
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The Navier–Stokes equations describe the behavior of fluids, such as liquids, and are therefore highly important and can be used to describe a range of phenomena. You can read about them in our post, which presents a physics-centric perspective on the subject and reviews the state of research at the time it was written, in 2021 [1].
The Navier–Stokes equations are included in the list of seven “Millennium Problems” defined by the Clay Mathematics Institute in 2000. Solving any one of the seven problems carries a prize of one million dollars [2]. In the case of the Navier–Stokes equations, the prize will be awarded for solving at least one of four questions concerning specific cases of the equations, and now two of them have been solved.
The equations describe flow over time, i.e., they represent the fluid’s behavior over time according to a particular initial state, which includes the velocity at every point. Until now, it was known that in three dimensions, for every initial state, a solution to the equations can be found that remains valid for a finite period of time. However, it was not known whether a solution always exists that remains valid “indefinitely”. The result now published shows that in three dimensions, there are initial conditions for which the equations have no solution for infinite time.
According to an OpenAI report from September 8, 2026, the model that solved the problem is an internal company model that is not yet available to the general public [3]. The problem itself was solved using a combination of ten thousand AI agents, over approximately 88 hours. A further 17 hours were devoted to producing a formal proof and verifying its correctness with a computer. The company published the formal proof, but as of the time of writing, it has not released the model’s full “reasoning” process that led to the solution.
And now for the not-so-scientific part: alongside this breakthrough, questions have arisen concerning credit for the mathematical progress. A rumor recently spread through the mathematics community that a solution had been found to one of the Navier–Stokes problems proposed for the prize, but no official information had been published. There was even a rumor that such a solution had been found by Anthropic. In light of this, according to the announcement [3], OpenAI decided to test whether its internal model could tackle one of these problems, or similar but easier problems. After successfully addressing a similar but less general problem, it moved on to the Navier–Stokes problems.
At the same time, shortly before OpenAI’s official announcement, two mathematicians, Tristan Buckmaster and Levent Alpöge, posted their progress on these problems on Mastodon, and also credited additional mathematicians [4]. According to the post, they solved, among other things, the less general problem—the one on which OpenAI’s model had practiced. This is a tremendous achievement and a step toward solving the corresponding Navier–Stokes problems, but it does not qualify for the monetary prize.
Among other tools, the two mathematicians used OpenAI tools for their research, raising concerns that OpenAI may in fact have used information generated through their work [5]. The company maintains that no direct use was made of this information, but it cannot rule out the possibility that information from the mathematicians’ work entered the training process of the model that solved the problem. Could such pieces of information, generated during the mathematicians’ work and incorporated into the model’s training, have helped it solve the mathematical problem? For now, the question remains open.
The current breakthrough joins a recently emerging list of open mathematical problems solved by “artificial intelligence” [6]—a development that raises philosophical, ethical, and practical questions about the practice of mathematics in the age of AI.
Hebrew editing: Smadar Raban
English editing: Elee Shimshoni
References:
- Post about the Navier–Stokes equations
- Clay Mathematics Institute publication presenting the “Millennium Problems” concerning the Navier–Stokes equations
- OpenAI’s announcement on Navier–Stokes
- Tristan Buckmaster’s post regarding the progress of his and Alpöge’s research
- Tristan Buckmaster’s statement
- Post about an AI disproof of an Erdős conjecture
Holds a BSc and an MSc degree in Mathematics from Tel Aviv University, and works as a data scientist. She has a broad interest in science, a passion for music, and a love for proverbs in various languages. She also serves as a scientific editor at “Little, Big Science.”
