But whether the equations always produce viable solutions remains the million-dollar question. 

To try answering it, Gómez-Serrano and his colleagues are using a very particular form of AI known as physics-informed neural networks or PINNs. Neural networks — lattices of computing nodes that function a bit like neurons in the human brain — are the bedrock of modern AI. But PINNs, which were developed at Brown by Professor of Applied Mathematics George Karniadakis, add a twist.

Standard neural networks are a true blank slate. They learn to recognize patterns from the ground up, with zero prior knowledge of any of the problems they’re used to solve. But PINNs start hard-wired with physical laws as they’re currently known. In this case, Gómez-Serrano is using PINNs that come embedded with fluid dynamics equations already on-board. That prior knowledge assures that any analysis done by the PINNs is firmly rooted in those known physical laws. 

The specific question Gómez-Serrano is wrestling with is whether there are any starting conditions for a fluid system that cause the equations to blow up as the fluid motion evolves forward in time. To demonstrate that PINNs have the potential to answer that question, Gómez-Serrano and his colleagues have been using them to look for singularities in equations that are related to Navier-Stokes, but simpler. The related equations operate in only one or two dimensions, which makes the problem a bit more tractable than in the 3D Navier-Stokes regime. 

Their work so far is promising. Last year, the team published a paper showing that PINNs had successfully found a few known singularities in the simplified equations, along with a few new singularities that no one had seen before. Importantly, the new singularities were “unstable,” meaning they arise only from infinitely precise starting conditions. If PINNs were able to find those vanishingly rare instances in the simpler equations, there is hope that they can eventually do the same for the Navier-Stokes equations. 

“We’ve shown that for these related equations, this AI approach was more successful in detecting candidates than any traditional method used before,” Gómez-Serrano said. “So, from the point of view of the performance, they were doing a lot better at the level of the discovery, meaning that they were able to find things that nobody knew existed before.”

Going from lower dimensions to full 3D adds significant complexity, and he doesn’t expect to have any definitive answers to the Millennium Prize questions overnight. And if the AI technique does identify a new Navier-Stokes singularity candidate, there will still be many hours of good old-fashioned mathematics work to prove that the AI is correct. That could take years. 

“There are far easier ways to make a million dollars,” Gómez-Serrano joked. 

But money isn’t his motivation. 

“For me this is just about understanding how the world works,” he said. “Do our equations fully explain the world, or are they just approximations? And what sort of match or mismatch is there between natural phenomena that we see every day and the idealized version we see through the mathematical equations?”

If AI provides a potential way forward, Gómez-Serrano is on board. 

Language models take on math

Gómez-Serrano’s work with AI isn’t limited to PINNs or the Navier-Stokes equations. Recently, he’s been collaborating with other mathematicians and a team of researchers at Google’s AI lab, DeepMind, on a project called AlphaEvolve. 

AlphaEvolve is an agentic AI system powered by Google’s Gemini large language model. It takes existing algorithms and codes — say, approaches to designing microchips or strategies for solving complex math problems — and makes a sequence of small changes to them, testing after each iteration to see if any changes improve the solution to the problem at hand. 

“It’s a cycle where the LLM generates the mutated code, and then through a scoring function we select the best answers,” Gómez-Serrano said. “This iterates over and over. Using this type of search, it was able to do very well in finding optimizers or mathematical constructions that were beating in some cases the previous literature.”

For a study soon to be published in Proceedings of the National Academy of Sciences, Gómez-Serrano and colleagues turned AlphaEvolve loose on 67 complex mathematical problems spanning mathematical analysis, combinatorics, geometry and number theory. For most of the problems, the system was able to independently discover the best-known solutions, matching the prowess of human mathematicians. And in a few cases, it was able to improve upon the best-known solutions. 

What was most remarkable about the work, Gómez-Serrano says, is the speed at which it was done. The team would start the agent working in the morning, and by the next day it would have a new solution. 

“We were solving or making progress on about one problem every other day, roughly,” he said. “That’s the pace we were at eight months ago, but that’s actually fairly slow compared to now. But back then, it was unprecedented in mathematics for anyone to make progress at that pace.”

That’s not to say that humans have been replaced in the math world, however. People still need to put in the sweat to rigorously check the output of the AI models, Gómez-Serrano says. And while there are those who remain skeptical, he believes that human and machine intelligence, working side by side, have a chance to make dramatic new discoveries. 

“There is a lot of space here to explore, a lot to pioneer,” he said. “And why not? It feels like there are many opportunities out there for whoever can manage to understand these tools and use them effectively.”