Did OpenAI solve the wrong Navier-Stokes problem?

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ttomjakubowski 1 day ago 51 commentsRead Article on scientificamerican.com

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jonlong•about 9 hours ago

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)!

bmacho•about 21 hours 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.

HarHarVeryFunny•about 9 hours 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 least interesting way possible.

tomjakubowski•about 11 hours 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

cpard•about 6 hours ago

When the news spread about the solution of this problem by AI we started wondering what will happen when AI will start generating proofs we can’t comprehend.

Today, we are discussing if AI cheated by picking the easy problem to solve which means that we at least still comprehend what’s going on.

I wish mathematics and the rest of the human intellect wouldn’t turn into content marketing that is generated primarily to trigger strong human emotions.

I feel that this is going to hurt both AI and the disciplines that can benefit the most from it

ChickeNES•about 10 hours 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?

alkyon•about 8 hours 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_aix•about 8 hours 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?

dgellow•about 9 hours 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

hatthew•about 9 hours 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.

dragonwriter•about 9 hours ago

Fefferman and the larger math community are not the same actor; a divergence between their concerns is not surprising.

ChickeNES•about 9 hours ago

yes!

ChickeNES•about 9 hours 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.

dragonwriter•about 9 hours 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.

yababa_y•about 9 hours ago

I will admit that when I heard they only had forced blowup I went back to sleep (but I've never been more that cursorily interested in analysis).

emmelaich•about 7 hours ago

https://archive.is/52HlJ

kittikitti•about 2 hours 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.

jongjong•about 6 hours ago

This situation illustrates exactly the limitation of AI and why we still need humans in the loop.

It reminds me of a junior coding bootcamp lecture I once gave many years ago before AI coding. One of the first slides said "Computers will do exactly what you say, not what you mean."

dist-epoch•about 6 hours ago

Seems to me this illustrates more the limitation of humans, they are the ones which chose the wrong problem, not the one mathematicians cared about.

ex-aws-dude•about 8 hours ago

I dunno this sounds like some insane goalpost moving

yieldcrv•about 9 hours ago

treating LLM's under a separate apartness ruleset isn't going to go well

ActorNightly•about 10 hours ago

Betteridge's law of headlines

dgellow•about 9 hours ago

That would be wrong here…

IshKebab•about 10 hours ago

The goalposts are moving so fast you can barely see them!

Next year's headline: But did Skynet kill all humans?

gr_norm•about 10 hours ago

This is mathematicians using the solution, explicitly acknowledged to solve the original formulation of the problem, to pose interesting new questions. That is what mathematics is.

Only someone who has never interacted with mathematics outside a rote-problem-solving capacity would describe it as you have.

IshKebab•about 9 hours ago

Please. The headline is clickbait, trying to imply that OpenAI somehow screwed up.

gr_norm•about 9 hours ago

What I've said is rather clearly spelled out in the article. Consider reading the it next time?

> It did, however, unambiguously solve the problem according to the Clay Institute’s original formulation.

dgellow•about 10 hours ago

That has nothing with moving a goalpost, it’s about understanding the actual result and digging into the details. Obviously when you do that things become more nuanced than at first glance

olliepro•about 9 hours ago

Classic example of moving the goal posts. “Exploiting a loophole” is how you solve many great problems in math.

dgellow•about 9 hours ago

That’s not at all the topic of discussion. The article is talking about the fact that OpenAI solved a version of the problem that is niche and isn’t the one the math community cares about

krackers•about 9 hours ago

>and isn’t the one the math community cares about

Then why was it allowed as an option in the millennium prize statement?

SpicyLemonZest•about 8 hours ago

At the time that the Millenium Prize problems were formulated, the force term was understood to make the problem more realistic, since real fluids are always going to have external forces applied to them. A blowup that happens under constant gravity, for example, would probably be no less interesting than an entirely unforced blowup. The strategy of constructing impossibly complex external forces to induce a blowup was pioneered by Córdoba and Martínez-Zoroa only over the past few years.