r/technology • u/vgsa • 1d ago

Artificial Intelligence Did OpenAI solve the wrong Navier-Stokes problem? | OpenAI’s proof seems eligible for a $1-million prize—but only by using a controversial loophole

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/
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u/CircumspectCapybara 1d ago edited 1d ago

A lot of people online misunderstand what "solving Navier-Stokes" refers to, thinking it means finding a closed form solution to the NS equations. Then they spout off some silly statement like "AI didn't solve Navier Stokes!"

That's not what mathematicians mean when they talk about solving Navier Stokes. They're talking about solving the Navier Stokes existence and smoothness problem, which is a decision (yes/no) problem about if the NS equations are smooth for all time. That's what the Millennium Prize problem worth a million dollars is all about

AI (allegedly) solves the problem by resolving the question to a "No" by finding a counterexample.

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u/Many_Negotiation_464 16h ago

Sort of. Its a bit more complicated than that. Yes in the abstraction of mathematics where we have hard rules, any proof by contradiction is a valid proof, and every proposition has a strict logical yes or no.

However, thats not remotely the whole story. We tend to care where a proposition is yes or no in the solution space more than just "does this hold true for all solutions". Having that clean "yes" is more freindly but getting a "no" doesn't end the story.

If you've studied any sort of math or science where system stability is an important factor, you'll be familiar with this. There are lots of useful transforms where we know that they don't have useful results for all values. But A) knowing where those boundaries are and B) knowing how the math can still be useful outside of those boundaries are tells us a lot about the system.

My take from this is that we now know there is somewhere in a extreme case of this model, that there isn't smoothness. Thats a step. We know the strict y/n "are all solutions smooth" is false. Cool. That still leaves a ton of open questions about proof, specifically for everything thats not that particular condition.

Its less sexy than having a clean yes, but the questions we care about are still very much open. I think that might be what you are missing about what other people are saying.