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Navier-Stokes: fluids, artificial intelligence and controversy

OpenAI has published that it has solved the Navier-Stokes historical equations. As one of the biggest problems in the world of science, the news has been dusty.

Something amazing has happened in the world of mathematics and physics: OpenAI claims to have solved the Navier-Stokes historical equations. It is one of the biggest problems in the world of science, so if it is true, it will be huge. But the issue does not end there. A controversy has arisen around this issue, which (again, if it is true that they have been resolved) will mark a turning point in the relationship between artificial intelligence and us.

Let's go little by little. Claude Luis Navier and George Gabriel Stokes were, respectively, a physicist and a mathematician. And they have been very popular in the scientific world precisely because they proposed the Navier-Stokes equations, useful equations within the dynamics of Newtonian fluids. That is, equations that serve to calculate how the most common liquids or gases move.

These equations are used, for example, to calculate the dynamics of water or air. Associated with them are atmospheric air, many meteorological phenomena, the current of a river (for example, in the calculation of the risk of a bridge falling) or the aerodynamics of an aircraft, basic physical equations that affect our safety and practicality.

“If it’s true, it will bring a turning point in the relationship between artificial intelligence and us.”

The Navier-Stokes equations, however, are very complex and do not work well in all cases. And above all, when they work, calculating their solution is almost impossible in most real examples. It is the typical scenario of mathematics: instead of calculating the exact solution, mathematicians must invent tricks to approach the solution.

That is why the Navier-Stokes equations are on the list of the most complex mathematical problems of the millennium and, of course, whoever solves them will receive a very high prize. Money – $1,000,000 – and enormous popularity. And you know what else.

On this occasion, the Navier-Stokes equations are supposedly solved by the company OpenAI. But it was a very controversial issue. In fact, in the last three years, the power of artificial intelligence in the field of mathematics has increased enormously: from the wrong realization of a simple sum, through reasoning models, it has come to make mathematical advances in research during this period. It has been an exponential evolution and now supposedly has solved a millennial problem.

The problem has not been solved through the GPT6 model, the most advanced on the market, but through another more powerful model that OpenAI is working on, they say. There is no doubt that this model is much more powerful.

“The one who solves them will receive a great reward: money – 1,000,000 dollars – and a tremendous reputation.”

However, is it true that he has solved the Navier-Stokes equations? And if it has, how has artificial intelligence done it? This is the main reason for the controversy.

To begin with, two mathematicians – Tristan Buckmaster and Levent Alpöge – have denounced that OpenAI has plagiarized their work in making the resolution. That is, artificial intelligence has not been based on zero, but on the results obtained by both mathematicians. Yes, without their permission. Either way, the context is complicated.

The work of both mathematicians provided a solution to the Navier-Stokes equations, but without taking into account the viscosity of the fluid. In addition, to carry out this work, they used Codex, an artificial intelligence system of OpenAI used for programming. Not only that: one of them, Alpöge, works at Anthropic, a direct competitor of OpenAI. Alpöge points out that his work is not a project in Anthropic, but a personal collaboration with the mathematician Buckmaster. For its part, OpenAI decided to try to introduce the viscosity factor and presented on September 8 that the equations were solved.

In the OpenAI result, singularities are created mathematically. Aided by the force of viscosity, the rotational speed of the fluid becomes infinite, which means one that is constantly stretching in a vortex. That is, rotating structures in the form of spaghetti are formed in the fluid. It's complex.

“Is it true that an artificial intelligence system is capable of solving one of these great millennial problems on its own?”

The core of the conflict is therefore in plagiarism: Buckmaster and Alpöge had very advanced solutions to the Navier-Stokes equations using the OpenAI Codex system. And OpenAI received the rumor that this work was on its way, and they launched one of their most powerful models to finish the calculation, probably starting from the method of the two mathematicians and training the system from their data.

In the end, there could be an ethical problem in two directions. On the one hand, Alpöge, an employee of Anthropic, is using a system of competitors to perform the calculation. And, therefore, it is an OpenAI client. On the other hand, OpenAI has used the methodology and data of a client to perform its calculation. It's a very dark matter.

We don’t know what this story will bring, but for years it will be remembered what has happened with the Navier-Stokes equations. The question remains: is it true that an artificial intelligence system is capable of solving one of these great millennial problems on its own, or are the methods of scientists still essential for the machine to achieve the result?