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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.
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
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.
"AI solved it."
"No we almost solved it."
"No way, it was AI all alone"
"no you used our data for train..."
"Guys, Guys calm! You did not produce any useful results!"
I think this is dramatized to the point it’s talking past the article, the math community isn’t really making any of these arguments from what I can tell.
The discourse is (1) models are capable of making really impressive mathematical advances, usefulness is not in dispute, (2) the frontier AI companies aren’t being super transparent about information sources so it’s hard to know exactly how to evaluate the level of capability that was demonstrated, and (3) there are lots of kinds of math that is interesting and there are open questions about how to get there.
In particular this article highlights a particular open question I’ve seen discussed on HN before, which is that the particular proof strategy of finding a counterexample might be more amenable to RL than other strategies of proof that might be needed to resolve the other branches of the Navier Stokes problem (and probably other similar areas of math)
yeah i agree its dramatized, but the situation was quite dramatized by the parties involved as well. i just find it quite funny, that the perceived drama might play out like this now.
Is it really impressive, or just kind of interesting?
If they spent about 10 GWh solving the problem (was it solved?) then that is much much more than 500 lifetimes of a human brain working.
A specific version of the problem has been solved. But not the broader, harder version. The article explains that fairly clearly.
I’m very anti AI and OpenAI, and do think it’s a pretty interesting finding! Very likely not worth their spend, but interesting and novel nonetheless the less
That is the best response I've heard to this argument. Assuming the solution is correct, the fact it is not the most interesting solution that could have been solved is besides the point. The team at OpenAI did an incredible job solving the problem.
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?
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
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.
yes!
Fefferman and the larger math community are not the same actor; a divergence between their concerns is not surprising.
But Clay chose Fefferman to represent "the larger math community"? I'm calling out Clay/Fefferman here, not the community. Though the community also had over 25 years to contest the problem statement, and had no problem with it until now.
I think you are agreeing with me? My point is that "the larger math community" failed to set the bounds of the problem correctly.
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.
The goalposts are moving so fast you can barely see them!
Next year's headline: But did Skynet kill all humans?
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
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.
Please. The headline is clickbait, trying to imply that OpenAI somehow screwed up.
What I've said is rather clearly spelled out in the article. Consider reading the it next time?
Betteridge's law of headlines
That would be wrong here…
Classic example of moving the goal posts. “Exploiting a loophole” is how you solve many great problems in math.
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
Then why was it allowed as an option in the millennium prize statement?
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.
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)!
treating LLM's under a separate apartness ruleset isn't going to go well
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).