But in just the last few months, an AI from DeepMind has done what no human could, unlocking new solutions to fundamental equations that could revolutionize how we predict weather, design aircraft, and understand the universe itself.
For decades, the math behind this challenge has been in a deadlock. These problems are so infamous that they are listed as one of the seven Millennium Prize Problems, with a one-million-dollar prize for solving just one. Now, DeepMind's AI has not only crunched the numbers faster but also uncovered entirely new solutions that were later confirmed as correct by human mathematicians.
This groundbreaking discovery centers on the Navier-Stokes equations, which describe how fluids like air, water, and gas move. These partial differential equations (PDEs) are notoriously difficult to solve due to "singularities" or "blowups" — moments in the math where values like velocity or pressure theoretically spike to infinity. While these don't happen in real life, understanding them helps scientists find the limits of their models. For more than 150 years, no one has been able to prove whether these singularities exist in the three-dimensional Navier-Stokes system. Solving this is fundamental to fields ranging from weather prediction to naval engineering and even astrophysics.
A New Approach with AI
DeepMind took a fresh approach, moving beyond brute-force simulations. They used Graph Neural Networks and Physics Informed Neural Networks (PINNs). Unlike normal neural nets that learn from data, PINNs are trained directly on the equations themselves. Their output is constantly checked against the laws of physics, allowing them to minimize the gap between their predictions and what the equations demand.
This method led to a breakthrough: the machine learning model discovered entirely new families of singularities that had never been described before. Mathematicians from New York University, Brown, and Stanford later confirmed these findings, proving that the AI's discoveries were mathematically solid and not a computational error. This was a real, verifiable discovery.
The AI uncovered fascinating patterns within these "blowups." By plotting a number called lambda (which characterizes how fast a singularity happens) against the order of instability, the DeepMind system found a clear, surprising pattern. This was observed across multiple equations, including the Navier-Stokes. This regularity suggests a vast landscape of other singularities waiting to be discovered.
Accuracy and Collaboration
The singularities the AI found are "unstable," meaning they require incredibly precise conditions to occur. While they may seem fragile, finding them is a massive deal because they reveal deep truths about the structure of the equations. The level of accuracy achieved was mind-blowing, with errors so small they were compared to predicting the diameter of Earth with an error of just a few centimeters.
This level of precision was achieved using advanced machine learning techniques, allowing the AI to capture delicate solutions that traditional methods could not. The visualizations created by the researchers, which map out how these singularities evolve, offer detailed views that were previously inaccessible.
This discovery is being taken seriously because it was a collaborative effort. DeepMind worked with mathematicians and geophysicists from leading universities to verify the results, grounding the AI's findings in existing theory. The AI acted as a research partner, pointing human experts toward solutions they couldn't reach on their own. As one of the lead researchers, Yang Ji Wang, put it, they transformed AI from a problem-solving tool into a "discovery engine."
This breakthrough marks the beginning of a new era of "computer-assisted mathematics," where AI systems can explore complex landscapes of equations and flag potential solutions for human experts to verify.
Reported by Tert Slamy.