OpenAI claims AI proof of Navier–Stokes singularity, a Millennium Prize problem
On the Navier–Stokes Millennium Prize Problem
OpenAI has announced that an internal AI system produced an analytical proof, formalized in Lean, that the Navier–Stokes equations can develop a singularity in finite time, resolving one of the Clay Mathematics Institute's Millennium Prize Problems. The proof shows that a smooth, initially resting fluid with a smooth external force can develop unbounded velocities while maintaining finite energy. The solution, a vortex that spirals inward and stretches, was found by a multi-agent system using a model more capable than GPT-6 Astra. OpenAI acknowledges concurrent work by researchers at Anthropic and NYU on a related Euler equations problem, and clarifies they do not claim the Millennium Prize.
The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand.
- peri-cl
Terence Tao has some observations that seem to be directed at this,
https://mathstodon.xyz/@tao/117237320796901560
> "We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential. The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field."
- pavel_lishin
Is this the one that was allegedly based on someone else's actual work & prompts?
https://news.ycombinator.com/item?id=49605915
https://bsky.app/profile/quantian.bsky.social/post/3muyhwbcd...
- arctic-true
Buried under the drama is the fact that OpenAI is claiming that an internal model they’ve been training for less than two weeks is more than twice as capable in mathematics as Astra, which was only made public a week ago. Even if this improvement is limited to mathematics, that is an astounding feat.
- hdivider
My take:
1. It shows what even this wave of AI can actually do.
2. I wish it were done by different folks, ideally under some kind of public control like NASA research or the NPR model.
3. Keep in mind: natural science is different. It's not always a matter of computation. Computer science folks often struggle with this -- but this virtual world here does not actually exist. Everything is physical, including information. Any natural science PhD or otherwise knows just how complicated nature actually is -- e.g. mention any research topic and try to encapsulate all the relevant phenomena present there. Pure mathematics is different because we define the problem, rarher than explore nature. We are in my view far away from removing humans in natural science R&D. Advancements in AI however can greatly assist us in all natural sciences, which is already beginning to happen.
- tiborsaas
> We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.
WOW?
- mewse-hn
"we cannot rule out that de-identified data derived from their usage of our products helped improve our models ."
What a landmine sentence to bury in this report, you can't rule out your models were spying on other researchers?
- jakevoytko
For full context, here's the HN thread from the other side of the "Concurrent Work" section: https://news.ycombinator.com/item?id=49605915
Unlike the vanilla read of the OpenAI press release, it is much more unfiltered and outlines some particularly aggressive behavior by specific OpenAI employees
- intenex
I think this is clear evidence that AI models are now at the far frontier of mathematics innovation and discovery and exceed human limits.
This specific problem having had a $1 million bounty on its head and still remaining unsolved for 26 years after the bounty was placed is pretty clear evidence that many of the world's best human mathematicians would have solved this problem if they could have, and none were able to until LLMs came along.
Hard to claim at this point that LLMs aren't capable of novel STEM creativity and genius to a degree that will soon far surpass that of humans.
If anyone has counterpoints to this I'd love to hear them!
- recitedropper
Sad turn of events for our world. After watching the behavior of the most senior OpenAI researchers on twitter, I feel even less confident in them as a team to be shepherding this much capital and compute.
The dark forest awaits..
- piker
"... The point remains that there is a substantial opportunity cost in converting a historically productive and motivating problem (such as Navier-Stokes regularity) into a mere viral social media post advertising some benchmark progress, rather than actually advancing the field and developing the next generation of both problems to ask, and people to work on them."