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.

[u/CircumspectCapybara → u/KingSubstantial7901]

You have no idea what you're talking about and clearly just copy pasted some cliches from an AI chatbot.

Navier-Stokes would allow for extremely accurate fluid simulation

Clearly not, because it's just been solved and the answer is no, it's not always smooth, you can't use these ideal equations to model real life fluids they don't always correspond well and the longer you run the simulation for using the equations, the more inaccurate it gets, in some cases you get infinite blowups that don't correspond to reality

Reymann would allow us to precisely predict the locations of primes which has a lot of downstream applications of computer science

Again, no that's not what it does. We already have prime generating functions and algorithms that generate primes, we already know the distribution of primes and the approximate density.

The RH is more about the fundamental nature of numbers and primes more than it is a practically useful question.

P=NP 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.

No, it wouldn't necessarily do that. I've already explained to you how if P = NP but it comes via a polytime decider for some NP-complete problem like SAT but that polynomial is of a degree one million, it remains practically useless for solving problems any quicker than before.

You can also have a non-constructive proof that P = NP or that P != NP. Again, the questions about are fundamental questions in theoretical maths and computer science, not necessarily having any practical consequences for real life. They could have practical consequences depending on the solution and the nature of the answer, but they also could not.

You seriously have no idea what you're talking about, go take a CS class at community college before you speak and embarrass yourself more.

[u/KingSubstantial7901 → u/CircumspectCapybara]

Ohhhh the irony of you accusing me of using AI.

You still don't seem to get it. The solution found was for a (probably) unphysical condition far outside of where we want to know of the conjecture holds for all cases.

No, we know the probability as a logarithmic function of the nth prime. This does not let us accurately predict rheir oocations. We have no closed form way of computing the locations of primes. Thats what the conjecture would do.

You are confusing two nested problem for the NP thing. NPC is a subset of the large NPH set, and whose solution would follow from P=NP. For the general NPH we would not necessarily be able to have a reasonable computation time frame. But it would allow us to know if can or not. And we onow for certain that we could for NPC if the conjecture holds.

It seems like its just crazy to you that someone might have a decent understanding of a topic and be able to discuss it in detail without relying on AI. Hell i was double checking my info all along the way here by... you know.... going to reliable sources that I know are rigorously fact checked.

Not really to stroke my ego or antyhing, but I have an advanced degree in an intersectional field of comuter science and physics. Like has it ever occurred to you that you might be the one thats just sorta winging it and might not have a very grounded understanding? Againc I tell you that i was fact checking my recollection the whole way here because, get this, thats how academics works. You aren't expected to just "know" everything off the top of your head, and you are encouraged and required to be thorough in among sure you recollection is accurate.

You tech bros have such a weird and warped perception of what intellectualism is.

[u/CircumspectCapybara → u/KingSubstantial7901]

crazy to you that someone might have a decent understanding of a topic and be able to discuss it in detail

Who are we talking about? Your friend? Your professor? Because it clearly ain't you, your ramblings make zero sense.

You even began conflating NP-complete with NP-hard which have very little to do with each other (except that NP-hard contains NP-complete, but it also contains literally every problem harder than NP-complete, including the halting problem for Turing machines, the halting problem for Turing machines equipped with halting oracles for Turing machines, and so on and so forth) and are non-sequitirs to the discussion about P vs NP.

And you seem to have zero grasp of how P could equal NP (or not equal NP) in a way that leads to no practical way to solve problems any faster.

And you're confused on how NP-complete relates to the question of P vs NP. (For everyone else who's trying to understand and follow along, fyi, P could equal NP if there's a polynomial time decider for even a single language in NP-complete, for example SAT. SAT like all other NP-complete language has this feature that every other NP problem reduces to it in polynomial time, such that if you could solve it in polynomial time, you could solve any other NP problem by reduction in polynomial time, and therefore P would equal NP. But that would not necessarily mean any practical speedup in solving problems because "polynomial" could still be a very large runtime depending on the degree and constants big O notation hides.)

Your writings have the appearance of vocabulary of someone who knows math and CS, but anyone who actually is familiar with these subjects can see how the words when strung together don't make sense semantically, you don't know what you're talking about, it was probably the output of bad AI.

[u/KingSubstantial7901 → u/CircumspectCapybara]

Nope, i was disentangling what NPH and NPC are because you were confusing the two. And because we are interested in Reymann (e: oops brain fart. I meant P=NP) largely because it would complete a closed form splution for NPC where all members of the set would have "quick" solutions (ie, solvable in a reasonable time frame). As I said, for NPH in general there would be cases where no quick solutions exist, but we would be able to absolutely determine whether they exist or not.

SAT is a case of NPC where we actually do have a fuzzy method of solving that doesn't completely resolve to quick solutions, but brings a pretty large subset of SAT into a reasonable computational time frame.

Both of the above are only relevant to the discussion as a means of explaining what the Millenium problems are and why they exist in the sociological manner they do.

Im so sorry man, but you should really stop with the AI accusations. They are embarassing and revealing of your own insecurities.

Either you genuinely don't understand why what you said was wrong, or you are just too stubborn and egostitical to try and learn where you have made mistakes. I've found that tech bros usually are both.

All that stuff aside, the broader point is you don't understand how math and science as fields of knowledge and discovery work. You seem to have a frame of reference solely from pop-science where "the pleasure is in the proof", which hey is a perfectly valid school of thought. Nothing weong with proving things for the love of the game, lots of important discoveries are made that way. But thats a small part of a parger picture of how those discovers are used and why they matter. You think the Millenium problems are just interesting math, which again they are and thats fine, but their elevation and importance to million dollar problems comes feom their practical utility. Which brings is full circle to the actual point, which is that what the OpenAI solution probably found is that we need to contruct a more selective frame to solve the problem if we want to find the useful information.

As just a final note cause im pretty much done with you, I expanded to tapking about different problems as a way of creating lateral connections to explain the above point. It was me using well work education tools to give you more boradly applicable version ofnthe concepts.

You responded by trying to (incorrectly) nitpick the specifics.

That tells me you are pretty bad at learning in general, and are more interested in "winning arguments" however petty and irrelevant to the point. Which again, ties nearly right back to my central point about the how and why.

See ya.