Did OpenAI solve the wrong Navier-Stokes problem?

Did OpenAI solve the wrong Navier-Stokes problem?scientificamerican.com

OpenAI’s proof seems eligible for a $1-million prize—but only by using a controversial loophole

122 points•tomjakubowski•9 days ago•70 comments•

#solve #navier #openai #stokes #wrong #wrong navier #navier stokes #openai solve

70 comments

No, OpenAI did not solve the "wrong" Navier-Stokes problem. OpenAI did not solve the hardest version of the problem (unforced blow-up), but did give a solution to the Clay Millennium Prize Problem as written and understood, choosing the explicitly allowed forced option.

SciAm writes "in a sense, the LLM found and exploited a loophole in the framing of the question". This is pure sensationalism. Choosing option (C) (out of an explicit list of four options) is neither a "loophole" nor something "found by the LLM"; everyone involved knew this was the option they were pursuing.

With the grumbling out the way, there is some actual scientific content to the article: there's a strong argument that OpenAI's method will not extend to the unforced case, leaving our understanding of NS incomplete. This negative result is itself new and interesting (and predicated entirely on the solution found by OpenAI)!

HarHarVeryFunny8 days ago

As I understand it the "loophole", if you want to call it that, is that OpenAI's custom-designed forcing function was smooth, as the rules said it had to be, but was non-analytic, consisting of some construction of "compactly supported bump functions", meaning a mass of tiny little pushes at precise points of space and time to push a vortex into blowing up the math.

hotdog14928 days ago

Reminds me of Maxwell's Demon.

dist-epoch8 days ago

Or a better framing: Clay Institute chose the wrong Navier-Stokes problem for a Millennium Prize.

seanhunter8 days ago

Not really. All of the Navier-Stokes options in the Clay Institute formulation of the prize are real problems of significant interest. The history of this particular problem is of finding specific conditions under which we can get something to work that turn out not to generalize in ways that people don’t expect eg iirc (It’s been a while since I read about it) there was a solution found early-ish in the 20th century for the 2-D case that turned out not to generalize to N-d, N>2 case, there are special conditions under which the turbulent terms cancel out and you can get smooth flow, vortices etc.

It seems to me that formulating problems at the boundary of human knowledge precisely is always going to be challenging and situations are bound to occur where you look back with the benefit of hindsight and wish that you had posed the question slightly differently based on some knowledge you didn’t have at the time.

glimshe8 days ago

Yet it had remained unsolved until OpenAI's effort...

hn_throwaway_998 days ago

Totally agree. I wouldn't quite call it clickbait, but the article does this thing I find annoying where it puts the "sensationalist" framing at the beginning (the "loophole" quote you put), but then closer to the end fully admits that it wasn't really a loophole in any case:

> It did, however, unambiguously solve the problem according to the Clay Institute’s original formulation. The official problem statement, penned in 2000 by mathematician Charles Fefferman, offers an option called “C,” in which solutions are allowed to use an external force like OpenAI’s.

p-e-w8 days ago

> SciAm writes "in a sense, the LLM found and exploited a loophole in the framing of the question".

God, it’s embarrassing to read stuff like this. They’re making it seem as if everyone involved was either stupid or dishonest just so they can pretend they have a scoop here.

casey28 days ago

It's just moving the goalposts, this happens every time an AI solves a problem, doesn't matter if the goalposts were there for 26 years.

What's interesting is that there are a set of people who are "in charge" and can as they wish arbitrarily set the goalposts to the thing that they happen to be best at. While this might be satisfying for an Humanity vs AI narrative, it's concerning for an us vs them one. Are these people really special? or do they just change the rules of the game so that outsiders (human or AI) can't win.

If it was you or I that solved this problem our would our rewards stop at $1M? Or would we get authority? If the achievement earns that for an insider but becomes 'just a solved problem' when an outsider does it, what exactly is being rewarded?

cyanydeez8 days ago

AI, the new minority!

bmacho9 days ago

Article says that there are 2 formulations of the NS problem and both are interesting: one is about fluid behaviour with no external forces, other is about fluid behaviour with external forces.

For a counter-example the latter is easier since you can have a tricky external forcefield.

HarHarVeryFunny8 days ago

Right - OpenAI proved the forced blow-up case rather than the harder unforced one, with the Millenium Prize problem statement saying it would be awarded for either one.

The forced version is easier since you can custom design the force function to get the result (it doesn't have to be a realistic force like stirring), so getting the blow-up might be regarded just as much a function of your bespoke force function as of the fluid dynamics itself, which is apparently what OpenAI did, pushing the definition of the force function being "smooth" to it's limit.

So, it appears OpenAI did legitimately meet the Millenium Prize solution criteria, but in the most unrealistic, and therefore least interesting, way possible.

adriand8 days ago

My friend, who is a mathematician, sent this in our group chat:

The title is a bit misleading. The variant with a smooth forcing was one of the four valid variants in the Clay formulation. It is interesting to solve it. It is still an interesting and impressive result. The no force version is also interesting and remains unsolved. It isn’t reasonable to just dismiss the proof on the grounds that 26 years later we claim it was never that interesting. This is the first time I’ve seen this attitude.

tomjakubowski8 days ago

I learned from this charming lo-fi video that it is actually much easier to find singularities in the Navier-Stokes equation for compressible fluids (which is out of scope for the Millenium Prize problem). The first was found in 1998 by Zhouping Xin.

https://www.youtube.com/watch?v=4wEn9B7pDV4

ChickeNES8 days ago

So what they are saying is that humans, in this case Charles Fefferman (a math prodigy, going by his history), failed to specify the problem correctly?

dgellow8 days ago

No, that‘s not the issue. If you look at https://www.claymath.org/wp-content/uploads/2022/06/navierst..., second page, you will see an option C is one of the four that is asked to be solved. And that option is the one that allows for an external force, which is what OpenAI solved. There is no question that OpenAI solved what the Clay institute is looking for. But that specific option C isn’t what the larger math community cares about, it’s a pretty niche case

hatthew8 days ago

I think GP's point is that if the larger math community doesn't care about option C, Fefferman shouldn't have given that option in the first place.

aesthesia8 days ago

It is interesting that there's a gap between the "prove N–S existence" conditions, which assume no forcing term, and the "counterexample" conditions, which allow a nonzero forcing term. In theory both (A) and (C) could be true.

ChickeNES8 days ago

> There is no question that OpenAI solved what the Clay institute is looking for. But that specific option C isn’t what the larger math community cares about, it’s a pretty niche case

I think you are agreeing with me? My point is that "the larger math community" failed to set the bounds of the problem correctly.

alkyon8 days ago

Fefferman was right to include option C as someone could have come up with less contrived counterexample accompanied by some interesting theorems that actually shed light on the general case

perching_aix8 days ago

Doesn't the way OAI's proof is formulated essentially rule that out, in that as long as an alternative solution concerns option C, it'd have to live in this same solution space they carved out?

dragonwriter8 days ago

Yes. And more humans—in this case OpenAI researchers—similarly failed in choosing how to direct the AI tools.

And yet another set of humans—Open AI marketers—made an error in how they sold the result of the preceding errors.

But its not news that computers are mere tools and that any error blamed on a computer involves at least two human errors, one of which is blaming the computer instead of the human(s) responsible.

Its perhaps a bit less obvious that every thing for which credit is given to a computer involves at least one human error—that of crediting the computer—and certainly can be more amusing when it involves a bunch of human errors.

chmod7758 days ago

There might be more value in having LLMs systematically hunt for mistakes in existing and widely assumed correct math papers, or hunt for counter-examples to things thought proven. There's likely to be a couple mistakes hiding in the less well scrutinized corners of mathematics.

Maybe something interesting will fall over because of that, who knows?

kittikitti8 days ago

The open question is whether the Clay institute will award OpenAI or others the Millenium prize for this solution to the Navier Stokes problem. I speculate that they won't. OpenAI's solution is undergoing peer-review and the counterargument presented in this article changed my mind. By default, if the Clay institute hasn't officially recognized the solution as true or likely true, I can't make an assumption that it is. I hope to go through the proof and perhaps AI can help better understand it and any potential weaknesses.

Read the full thread on Hacker News →

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