Spanish mathematician Javier Gómez Serrano partners with Google DeepMind to tackle the Navier-Stokes million-dollar problem.

The mathematician Javier Gómez Serrano, born in Madrid 39 years ago, has teamed up with the artificial intelligence giant Google DeepMind to try to soon solve one of the most puzzle-like enigmas known, that of the Navier-Stokes equations, he reveals to Adolfo Kunjuk News. This is one of the seven Millennium Problems, for which the Clay Institute in the United States offers a reward of one million dollars (and “immortal fame,” as the Spanish researcher emphasizes). The so-called Navier-Stokes Operation, which has been ongoing for three years with a team of 20 people, has been conducted discreetly until now, although Google DeepMind’s head, Demis Hassabis, hinted in an interview in January that they are “close to solving one of the Millennium Problems,” without mentioning which. “We will see it in the next year or year and a half,” he stated.

Gómez Serrano, a professor at Brown University (USA), speaks publicly for the first time about the operation. “There is currently a general consensus in the community that the problem will be solved soon, but no one knows who will do it or how,” he explains. The challenge dates back to the 19th century when two mathematicians, the Frenchman Henri Navier and the Irishman George Gabriel Stokes, independently published, in 1822 and 1845, the equations that describe the movement of fluids, such as water and air. Based on temperature, viscosity, and the initial speed of the fluid, the equations calculate its speed at a later moment. Two centuries after their formulation, it is still unknown whether solutions always maintain regularity or if an explosion, a sudden change in behavior, can occur, as if a tsunami suddenly strikes in calm seas. These equations are essential for predicting relevant phenomena such as weather, catastrophic flooding, aircraft movement, or blood flow in humans.

The resolution of the enigma seems imminent. Gómez Serrano co-leads a team of five academics who met while working at Princeton University and are now scattered across institutions in the United States. They are two geophysicists—the Taiwanese Ching-Yao Lai and the Chinese Yongji Wang, authors of complex models to calculate melting in Antarctica—and three mathematicians: the Australian-British Tristan Buckmaster (the other co-lead), the Spanish Gonzalo Cao Labora, and Gómez Serrano himself, who grew up in the working-class neighborhood of Puente de Vallecas, where he often attended calimochero rock concerts at the legendary Hebe venue, like those of the band Porretas.

Great minds in mathematics have faltered while trying to solve this challenge, dedicating the best years of their academic lives only to find themselves trapped in a dead-end. In 2014, the team of Thomas Hou at the California Institute of Technology did achieve a significant breakthrough, thanks to a prior simplification of the problem. Hou’s group did not use the Navier-Stokes equations but an earlier version proposed in 1752 by the Swiss mathematician Leonhard Euler to describe the movement of ideal fluids, without viscosity. The researchers produced a simulation of a fluid within a cylinder, which, under certain initial conditions, seemed to result in a “singularity”: the sought-after tsunami in calm seas. Gómez Serrano’s team employed artificial intelligence techniques—machine learning neural networks—to refine the solution and understand where and how the singularity forms. Their results, published three years ago, were interpreted as a sign that the solution to the million-dollar problem was imminent.

The Spanish mathematician believes that there are only three other groups seriously competing to solve the enigma worldwide: the aforementioned Thomas Hou team in California; the tandem formed by the Egyptian Tarek Elgindi and the Italian Federico Pasqualotto, also in the United States; and the team of Diego Córdoba, a 53-year-old from Madrid who directed Gómez Serrano’s doctoral thesis over a decade ago at the Institute of Mathematical Sciences in Madrid, focusing on how waves break in the sea.

“The Navier-Stokes problem is tremendously difficult,” acknowledges the professor at Brown University. “People have not succeeded using traditional mathematics. The difference in our strategy, compared to everyone else’s so far, is the use of artificial intelligence. That is the advantage we have, and we believe it can work. I am optimistic; the progress is very, very rapid,” he notes. In his opinion, the solution will arrive at some point in the next five years.

Gómez Serrano has just participated in another historic advancement with Google DeepMind: AlphaEvolve, a new artificial intelligence system that solves complex mathematical problems with unprecedented effectiveness. The Spanish professor and his American colleague Terence Tao—considered the best living mathematician—trained the program for four months with a dozen puzzling challenges. “In 75% of cases it matches the best human result. In another 20%, it improves it. A success rate of 95% is frankly impressive,” notes Gómez Serrano.

“I believe that a trained human, reading the related literature, programming a lot, and preparing for several months, might achieve it. But AlphaEvolve did it in one day. That is really the advantage. It can become a tool that greatly accelerates research. It will change the way mathematics is done,” he argues.

The head of Google DeepMind, British neuroscientist Demis Hassabis, and his American colleague John Jumper won the Nobel Prize in Chemistry last year for creating AlphaFold2, an artificial intelligence system capable of predicting the intricate structure of the 200 million known proteins. The program accomplishes in minutes what previously required months of work. The revolution of the new AlphaEvolve system is that, unlike AlphaFold2 and the program designed for the Navier-Stokes enigma, it is not created with machine learning to solve a very specific issue, but is an extensive language model, like ChatGPT, that solves problems across a wide range of mathematics without being a specialist.

Demis Hassabis has predicted that the so-called general artificial intelligence, software with human-like intelligence and self-learning capability, will arrive around 2030. Gómez Serrano is more cautious. “There are people, bolder than I am, who do forecast that within 5 or 10 years, artificial intelligence will be at the level of the best mathematicians in history. I don’t know, but I know that it is progressing at an extremely rapid pace,” he reflects. “There are two currents: optimists and pessimists, who think of Terminator [the 1984 movie in which an artificial intelligence rebels against humans]. I believe that we will be asking more complex questions, that we will be able to better understand nature and design better materials and medicines. I believe it will change the world, and I want to believe it will change for the better.”