“OpenAI has perhaps solved the Navier-Stokes equations”, one of the so-called millennium problems, i.e. the 7 mathematical problems that are considered among the most difficult and important formulated by me. This is what the company that develops Chat-GPT declared on September 8th. If the resolution of the problem turns out to be correct, it would be a very important scientific achievement: the Navier-Stokes equations, in fact, are fundamental for understanding fluids and are used in many different fields, from the design of planes and ships to weather forecasts.
The announcement, however, is already at the center of some controversy within the scientific community: while OpenAI describes the achievement as a technological victory for AI, the mathematician Tristan Buckmaster, professor at New York University, stated that OpenAI would use some of its important discoveries on the topic to train its models and thus make them capable of solving the problem.
What are the millennium problems and the Navier-Stokes equations
First of all, what are the so-called “millennium problems”? These are 7 particularly complex mathematical problems, identified in 2000 (hence the name) for the solution of which the Clay Mathematic Institute – an important international mathematical institute – has offered a prize of one million dollars each. Until now, only one of the 7 problems – the one relating to the Poincaré Conjecture – had been solved by the Russian mathematician Grigori Perelman.
The problem that OpenAI may have solved concerns the “existence and regularity of the solutions of the Navier-Stokes equations”. These equations, formulated in the nineteenth century by Claude-Louis Navier and George Gabriel Stokes, serve to describe the movement of fluids, that is, gases and liquids. Simply put, the Navier-Stokes equations relate the speed, pressure, density and viscosity of a fluid at a certain moment, so as to be able to predict its progress in subsequent moments.
In engineering practice, and therefore in application to the real world, the Navier-Stokes equations work very well, providing excellent predictions confirmed by experience. But from a theoretical point of view, some aspects of their behavior are not yet fully understood. In short, there is no rigorous demonstration that the solutions always behave regularly.
So, what does the millennium problem say about these equations? The underlying question is very complex from a mathematical point of view, but – simplifying a lot – we ask ourselves if there is always a solution to these equations which, if it works at a certain instant, functions at any other time in the futurethat is, it is able to describe the velocity of the fluid without “going crazy”.
In mathematics, “always functioning” is called “not having singularity”, that is, not having borderline situations in which the equations “break” and stop giving “sensible” results, the so-called blow-up.
The example of coffee and infinite speed
Let’s take an example to understand what what we have described means. Let’s imagine taking a teaspoon and stirring our coffee (which is a fluid). The solution to the Navier-Stokes equations should tell us – starting from the zero instant in which the coffee was still and knowing how fast we turned the spoon – how fast every single drop of coffee moves, at every point in the cup, moment by moment.
On a practical level, we know very well that the coffee, after we mix it, will never reach an infinite speed, because infinite speed does not exist in reality. But in the mathematical world it exists.
So, our millennium problem asks whether there is a finite T value – like 5 seconds – which if we put it into our Navier-Stokes equation that describes the motion of our coffee, starting from our initial conditions, gives us the infinite result. Mathematically it could happen – the infinite value as a result is accepted – but in reality it is not.
In this scenario, the equation before that instant was faithful, regular, while after the instant T, the equation “broke”, is no longer able to provide a result.
The solution and the use of 10,000 AI agents
As we said, OpenAI declared that it used its most advanced internal model, not yet available to the general public, to solve the problem we have just described, obtaining the following result:
there are conditions under which an initially still and regular fluid can reach infinite speed under the action of a regular force.
In particular, the solution described concerns a vortex that twists on itself, lengthens and accelerates progressively, shrinking more and more, while its energy remains finite, as required by the laws of physics.
To achieve this result, OpenAI declared that it used various AI agents, i.e. artificial intelligences capable of carrying out autonomous or semi-autonomous actions coordinated with each other. The agents could execute code and access information available on the internet in a secure manner, and were based on a new ChatGPT model not yet released to the public.
The company says it initially tested the system on Euler’s equations, a simplified version of the Navier-Stokes equations, employing around one hundred agents for around fifty hours. After this first result, OpenAI moved its resources to the Navier-Stokes equations, employing around ten thousand agents in parallel.
According to the company, the solution was found 88 hours after starting the project, with computational costs amounting to millions of dollars. Despite this, in the event that the solution to the millennium problem is confirmed as correct, OpenAI has declared that it does not intend to request the million dollars foreseen by the prize.
This result will now have to be subjected to an independent verification process by the scientific community. In this case, however, the verification could be more delicate than usual. Just as OpenAI published its result, two researchers who were working on the same problem said they had arrived at a partial solution and raised suspicions that the company may have used their conversations with the AI to train the model that addressed the problem.
OpenAI is accused of “copying” the demonstrations of two users
The day before OpenAI’s announcement, New York University mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge made public some partial results on the Navier-Stokes problem they had been working on for months.
The results had been obtained with the help of several AI models, including Claude, developed by Anthropic, and Codex, developed by OpenAI. In addition to these results, Buckmaster published a statement in which he tells his version of the facts: OpenAI started working on the problem only after hearing some rumors that Anthropic, the company Alpöge works for, was close to solving the millennium problem. Hence the suspicion of the two researchers that OpenAI began training the new model after learning of the news and that, to do so, it used the conversations that the two had had with Codex.
In his reconstruction, Buckmaster also says that he contacted OpenAI to discuss the situation and received two proposals. The first involved a joint announcement about the discovery, with the publication of his work followed a day later by that of OpenAI. The second consisted of a single article by Buckmaster on the result relating to the Navier Stokes equations, in which the contribution of the OpenAI model would also be recognized. In both cases, however, Buckmaster should have excluded Alpöge, who works for a competing company, from the job.
OpenAI, for its part, responded to the accusations by declaring that it had not had access to the work of the two researchers before its publication and, in particular, that it had not used specific user data to solve the problem. However, the company clarified that it cannot exclude that anonymized data, derived from the use of its products in general, contributed to the improvement of its models.
