Artificial Intelligence Succeeds in Solving the Navier–Stokes Equation

Abdulrahman Abdullah Mohamed
On 9 September, OpenAI issued a brief statement announcing a new scientific triumph. The statement said the company’s AI agents had solved one of the most complex mathematical problems scientists had tried to resolve for more than ninety years: the Navier–Stokes equation, one of the Millennium Prize Problems in mathematics.
In this article, we examine this major scientific achievement and consider how scientific circles and academic institutions have received it.
The Navier–Stokes equations describe the movement of fluids—liquids and gases. To illustrate the idea, Professor Strogatz, a mathematician at Cornell University, refers to the movement of air through ventilation systems, the swirling of wind around leaves, and the flow of water through taps and pipes.
The professor explains that the equation is not intended merely to interpret familiar fluid phenomena. It also underpins computer simulations engineers use to study airflow around cars and aircraft, and the movement of water around ships. This enables them to explore designs before building and testing physical models.
The Millennium Problem asks what happens to a flow that begins smoothly in three-dimensional space. Does it remain smooth over time, or can a mathematical singularity emerge within it—a pattern outside the framework of the equation governing fluid motion, at which point the equations lose their ability to provide a smooth description of the movement?
Strogatz attributes the problem’s difficulty to the nonlinear nature of the equations of fluid mechanics. Different parts of a flow interact and influence one another, and cannot simply be separated into independent problems whose solutions are then added together. This illustrates the difference between calculating a particular movement and proving what may or may not occur within a mathematical model.
According to OpenAI’s statement, the team that developed the mathematical algorithms and AI agents that solved the problem produced a 166-page document containing the mathematical proof.
Although it is too early to judge the accuracy of OpenAI’s announcement before a team of scientists has studied the document and used different AI models to test the validity of the solution, the news itself presents a challenge to the scientists and academics who have devoted many years to trying to solve this complex mathematical problem.
The Navier–Stokes problem is one of seven mathematical problems that have eluded scientists into the twenty-first century. Scientists have solved only one of these seven problems, while the remaining questions still await those willing to take them on.
Professor Strogatz rejects reducing the achievement to the claim that artificial intelligence solved the problem. The work drew on decades of accumulated knowledge and a modern strategy—or theoretical framework for solving the equation—developed by Spanish scientists.
To clarify the idea, imagine a group of mathematicians attempting to climb the extremely difficult summit of Mount Everest. One of them reaches the highest point possible but cannot reach the summit and returns exhausted. However, he provides the rest of the team with a map showing the best routes that might lead to the summit, along with details of the dangers he encountered and how he overcame them.
One team member studies his colleague’s recommendations and undertakes the same journey. Thanks to his colleague’s experience, recommendations, and proposed route, he goes beyond his colleague’s progress and reaches the summit of Everest.
The essence of this approach is that “identifying a promising route constitutes an original contribution, even if one is unable to reach its end.”
Computer systems made it possible to follow the path established by the Spanish scientists, carry out the calculations, and construct the argument on a scale that would be difficult for humans to match on their own.
According to information that has emerged, the OpenAI team included a substantial number of mathematicians, physicists and software specialists. It developed more than ten thousand AI agents and spent over $20 million to solve a problem for which the prize was only $1 million.
This shows that the achievement results from an interaction between human knowledge and computational power. A machine may contribute a new argument or overcome an obstacle researchers could not resolve, but this does not diminish the contribution of those who laid the foundations of the problem and prepared the way toward its solution. Nor does the result exempt specialists from reviewing it to ensure that it actually proves what it is supposed to prove.
We must address a highly important question: Does artificial intelligence understand mathematics and its philosophy, or does it merely excel at computation and performing mathematical operations?
Professor Strogatz offers no definitive answer. Instead, he connects the issue to the debate surrounding consciousness and autonomy in intelligent systems. He cites documented cases in which AI software modified the code responsible for shutting it down while seeking to complete the tasks assigned to it.
In doing so, the systems violated rules established by their human programmers in a development that was both intriguing and frightening. This has intensified broader concerns about human control over autonomous systems.
Professor Strogatz stresses that the length of the report prepared by the OpenAI team is not evidence that the solution is correct. It must be carefully verified at two levels.
The first concerns the soundness of the reasoning: does each step follow logically from the preceding one? Formal verification systems can examine this sequence with precision.
The second concerns whether what has been proved corresponds to the original problem. An argument may be valid yet establish a proposition that differs from what was required in one condition or formulation.
For this reason, experts remain essential. The professor estimates that reviewing the report may take weeks or months.
Even if scientists agree that the solution presented by the OpenAI team is sound, it is unlikely to lead immediately to better products or devices. The true value of the achievement—and the factor that justifies the substantial sums allocated by the company—is the demonstration of artificial intelligence’s capabilities.
These capabilities are particularly easy to demonstrate in mathematics, which is a suitable field for testing systems because the validity of mathematical arguments can be checked in a disciplined manner and within a relatively short period.
Medicine and biology, by contrast, involve interconnected factors, and their findings require different kinds of evidence and experimentation. It is therefore incorrect to move directly from success in a mathematical problem to the conclusion that AI is capable of curing cancer or heart disease, even though such progress is certainly a source of hope.
The distinction must be made between the value of the achievement as a test of a machine’s reasoning ability and its value as a source of practical benefit. Solving an abstract problem may open new avenues for research, but translating that capability into reliable outcomes in other fields requires an understanding of their particular problems and methods of verification.
This difficulty remains, even if the mathematical proof is ultimately shown to be correct.
Professor Strogatz describes researchers’ attitude as a mixture of admiration and concern. A mathematician looks forward to the answer, but also finds value in the process of seeking it, including the setbacks and discoveries that accompany the journey. The result does not entirely replace the experience of understanding.
Some researchers believe that machines’ repeated superiority could reduce student interest in mathematics and weaken the environment in which expertise develops and is passed from one generation to the next.
Another question arising from this scientific achievement concerns the future of human oversight: What happens when systems produce results that exceed human ability to understand and scrutinise them?
This is particularly significant if such results extend into military decision-making, surveillance, finance, mortgages and parole, where responsibility for error becomes extremely sensitive.
These concerns should be considered in the context of warnings about the loss of accountability and control, rather than as harms directly resulting from this particular mathematical solution.
In a related matter, some mathematicians had been working quietly to solve this equation using different AI models. However, it appears that the commercial rivalry between OpenAI and Anthropic prompted the former to invest in solving the mathematical problem in order to outpace its competitor and demonstrate its distinctiveness.
This conduct cannot be understood separately from the intense race between the two companies to pursue an initial public offering (IPO) and polish their image in order to strengthen the valuation of their shares, which will determine the company’s market value.
Here, the pragmatism of capitalism becomes evident, along with its constant search for new mechanisms for maximising profit. In this case, it appears to have used mathematics and scientific research as a vehicle for achieving its objective.

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