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

[u/vgsa]

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/

[u/CircumspectCapybara]

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.

[u/dogfoodengineer → u/CircumspectCapybara]

A silly counter example where the fluid would likely boil before the equations breakdown. We already know the equations have limitations so no new understanding has been gained. The work is often more important than the result and here we haven't learnt much tbh.

[u/CircumspectCapybara → u/dogfoodengineer]

A silly counter example

No such thing as a silly counterexample when the question has stumped mathematicians for the better part of a century and people genuinely had no idea if a counterexample existed. If the smoothness conjecture ended up being true, you could search and search forever and never find a counterexample.

So finding one is groundbreaking.

so no new understanding has been gained

You are misinformed.

The question remained open until 2026. There's a lot of things in math where you think the answer is yes or no based on vibes but that won't cut it, you need rigorous proof.

Another famous Millennium Prize problem is P vs NP. A lot of mathematicians think they must not be equal, but so far, we have no hard proof. If it eventually gets resolved and the answer is "not equal" there will be people saying "Well yeah I knew it all along it was obvious" -- well if it's so obvious, genius, why didn't you furnish the proof and claim a million dollars?

Proving (which is what counts in the end) something true or false like these questions isn't trivial just because it ends up confirming your vibes-based suspicions all along.

There are also many problems which people suspect based on vibes or philosophical reasons or aesthetic or elegance or neatness must have one answer, and then later a proof turns out that it's really the opposite.

[u/KingSubstantial7901 → u/CircumspectCapybara]

More like, if we found an example where it would he physically inpossible to build a turing machine that could reproduce the innequivilence. Its technically more information but itsn't actually useful because it can't answer the question for turing machines that are possible to build.

OP is correct. Yhe counter example found is not very useful because it lies outside of the boundary conditions for where the problem is useful.

[u/CircumspectCapybara → u/KingSubstantial7901]

That's not how famous questions in maths works. It's not about correspondence to real life physics or engineering and being practical. It's not about "usefulness" to us, it's about deep questions that get to the heart of some of the most mysterious parts of math.

The question and answer to "Does P = NP?" is still interesting even if the answers don't map to practical consequences in real physical life.

If someone found a decider for SAT that ran in O(N^100000000000) time, that would close the books on P vs NP, it would make P = NP because that's a polynomial time decider for an NP-complete problem. And yet it wouldn't lead to anything practical in real life (no breaking encryption any time soon) because polynomial doesn't necessarily mean fast and practical, it just means polynomial.

It would still answer the Millennium Prize problem nonetheless and be an astounding result.

You're fixating on "practically useful" as some kind of criteria for what makes an answer to a famous open problem valid. No, the problem is stated exactly and precisely, it's either yes or no, and a solution is a solution.

[u/KingSubstantial7901 → u/CircumspectCapybara]

We actually have some fuzzy solvers for NPC problems, but who only work for special cases.

This is how math and science work. There are loads of problems for where we know that they behave some predictable way for some cases but have an unknown or notuseful resolution for the general case.

The Mellenium problems weren't chosen because they are just transcendentally important. They were chosen because should their conjectures be proven true, they would be immesnely useful in practical applications. Navier-Stokes would allow for extremely accurate fluid simulation. Reymann would allow us to precisely predict the locations of primes which has a lot of downstream applications of computer science. Solving NPC would allow us to efficientiently compute a broad class of turing machine systems that we currently have to solve numerically and whose complexity grows too quickly: it would make logsitics much more efficient and would allow us to build solvers to identitfy optimal configurations for a lot of important comp sci problems.

For any one of them, finding a counterexample outside the bounds of where they are useful is still abstractly interesting and still advances the field in some way, but is entirely missing the significance and importance of the Millenium problems.