A.I. Solves the Navier-Stokes Millennium Problem

By Anika O. '30

On September 5th, 10,000 internal OpenAI models solved a problem that the greatest mathematical minds of the century had struggled with, completing it in just 88 hours. By the end of the day, the entire scientific community was in existential crisis mode. But should we be, and what does this move of AI brilliance really mean for math going forward?

Background

The Navier-Stokes problem is one of the seven math problems named by the Clay Mathematics Institute as Millennium Prize Problems. These problems are thought of by the mathematical community as sitting at the frontier of human understanding, and a solution to one would potentially be revolutionary in opening up new branches of both thought and real-life utilization. The founders, Claude-Louis Navier and George Gabriel Stokes, initially developed the Navier-Stokes equations to describe how fluids move using Newton’s Second Law (F=ma). In particular, the equations track how continuous fluids move, rather than individual molecules. An open question for mathematicians has involved investigating whether a fluid that begins its motion smoothly can develop a singularity. A singularity means that the speed of the fluid grows infinitely large within a finite amount of time. AI proved that an initially smooth fluid can develop such a singularity.

The Controversy

Just 12 hours before OpenAI announced its proof, mathematicians Tristan Buckmaster (NYU) and Levent Alpöge (a researcher at Anthropic) stated that they had been working on a special case of the problem for a while, and their work-in-progress may have been leaked to OpenAI a few days prior. Specifically, Buckmaster noted in a document posted on his website, “I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.” In their announcement of the breakthrough, OpenAI stated: “We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed to solve this problem.” In addition, OpenAI said that there were “significant” differences between the two proofs. However, in a September 8th tweet, OpenAI wrote: “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”

The work-in-progress had been stored in an OpenAI Codex model, which could potentially make it accessible to the OpenAI team. This potential breach raises a significant question of user privacy within the AI world.

Looking Forward and the Broader Community

Recently, it has become more and more apparent that math has grown into something of a source of competition for various AI agents. But if this continues, what effects could it have on our perception of science as a whole? 25 Fields Medalists created and signed a document responding to these rapid developments. This document can be found at https://mathandai.org/ and in a blog repost by Terence Tao (a Fields Medalist and professor at UCLA), where you can also find other interesting posts about AI and math.

In summary, the declaration discusses the necessary idea that “as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place”. In particular, the mathematicians emphasize that the reason mathematical research is done is to gain a better conceptual understanding of math as a whole, and to plant seeds of new insightful ideas. Famous problems, they say, can be thought of as lighthouses, which can be useful in measuring how much our understanding of the landscape of math has improved. However, spitting out solutions to these problems is not the end goal. In the same document, they write, “Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.”

Looking forward, there is only one thing we can say for certain: AI will forever transform our society, not only in scientific study, but more broadly, even our day-to-day life. So, as students, what can we do to make sure these changes will have a positive effect on society?