Artificial intelligence solved the giant challenge of mathematics, solved the equation of liquids and gases

It is the solution of the Navier-Stokes group of equations for the behavior of liquids and gases. He found a mathematical solution of the OpenAI company artificial intelligence.

“Artificial intelligence has entered mathematics strongly this summer. The role of mathematicians is changing,” says the professor Mikko Salo from the University of Jyväskylä.

The solution of the Navier-Stokes group of equations has been sought for nearly a hundred years. It is one of the seven so-called Millennium Problems which Clay Research Institute announced in 2000. The solver is promised a million dollars.

Navier-Stokesin the group of equations describes how the speed, pressure, density and viscosity of the flowing liquid or gas, i.e. internal friction, are related to each other.

Equations used all the time for example, to model the weather, sea currents, the flow of liquid in a pipe or without movement around an airplane wing. They are used, for example, in the design of cars and aircraft or in the design of power plants.

They are therefore widely used, and computers provide usable results for flows.

Clean however, mathematically they are very difficult to solve. Mathematically, it is a time-dependent partial differential equation puzzle.

“The group of equations contains two descriptions of the phenomenon that competes against each other in flows. It is the driving force of the flow that is resisted by viscosity, or stickiness,” Salo describes.

The moving effect can cause complex eddies, such as are created when cream is mixed with coffee, for example. Viscosity tends to equalize flows, as for example when the temperature equalizes in a room.

The mathematical goal in the case of Navier-Stokes is to investigate whether the equations always have a solution in theory and whether the solution is, so to speak, smooth. Smoothness refers to the fact that the properties of a liquid or gas change without jumps.

“There are two options. Either you have to show that eddies are created in the flow, the speed of which accelerates to infinity in a finite time. The other option is to show the opposite, that such eddies cannot be created,” Salo simplifies.

The equations deal with these so-called fluideja as a uniform medium and not as individual molecules.

The solution is complicated by the non-linearity of the equations. The speed of the flow affects how the speed changes. A small change can change the subsequent flow in a way that cannot be explained simply by adding up the individual effects.

Salo points out that OpenAI’s result is still very fresh and its evaluation is just beginning.

Artificial intelligence models use learning neural networks. They are, in a way, models of information processing imitating the human brain, which utilize a huge amount of digital information.

“Artificial intelligence has been taught using the results of experienced researchers. This solution is based on Spanish mathematicians Diego Cordoban and Luis Martinez-Zoroan for research from 2023”, says Salo.

Essential was the use of the programming language Lean in the review, says the computer science professor Teemu Roos from the University of Helsinki. Lean was made just for mathematicians.

“That’s absolutely the key question. There is always the possibility of mistakes in the mathematics that people do. Lean writes in the form of a proof, where it can be automatically and seamlessly checked,” says Roos.

“This is important because artificial intelligence systems make mistakes, i.e. hallucinate even more than humans. In addition, no human can check the certificates made by artificial intelligence due to their length,” Roos continues.

OpenAI: n the group used the company’s in-house AI model and likely several million dollars worth of computing power. The solution used 10,000 artificial intelligence agents that sent each other about three million messages.

“This is a new qualitative feature. The artificial intelligence agent has the ability to do more than respond to prompts made by the user. They can create their own program snippets and exchange information with each other,” Roos describes.

Currently, such computing resources and internal models are mostly used by artificial intelligence companies.

OpenAI is not going to require The million dollar prize promised by the Clay Institute.

Alongside OpenAI, another artificial intelligence company, Anthropic, has tried the Navier-Stokes solution. The company doubted at the beginning of September According to the BBCthat OpenAi has utilized Anthropic’s work.

The other research group has an Anthropic employee who uses OpenAI’s tool, and information could therefore have been leaked.

Equations is named after the French by Claude-Louis Navier (1785–1836) and of Irish birth George Gabriel Stokesin (1819–1903) according to They were prominent researchers in flow dynamics.