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Did OpenAI solve the wrong Navier-Stokes problem?

122 points · 70 comments · tomjakubowski

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bmacho · 1790070475 · focus · HN ↗
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.
1. tomjakubowski · 1790105633 · focus · HN ↗
  [delayed]
2. HarHarVeryFunny · 1790111673 · focus · HN ↗
  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.
  1. adriand · 1790117107 · focus · HN ↗
    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.
    pennomi · 1790174083 · focus · HN ↗
    It’s just people trying to cope in the most human way possible: “I don’t like AI, therefore AI is stupid, therefore its proof must be uninteresting.”
    Almost never do people judge situations entirely on merit.
    HarHarVeryFunny · 1790182959 · focus · HN ↗
    Not really - if you read beyond the headlines of "mathematicians don't like AI", and your own imagined (incorrect) reasons they may have for that "AI is stupid", then the truth seems a bit more interesting.
    If you want to abide by your own words and judge the situation on it's merit, then you need to look at the specifics, meaning the OpenAI proof itself (166 pages), and the analysis of it that is only just beginning. Assuming that the proof is wonderful and provides much insight into Navier-Stokes is just as dumb a take as saying that it doesn't. Judge it on its merit.
    The Scientific American article is sadly paywalled, but at least part of the discussion is based on the paper below, whose work the OpenAI proof appears to build upon.
    <a href="https://arxiv.org/pdf/2609.20803" rel="nofollow">https://arxiv.org/pdf/2609.20803
    When discussing the proof, below, with Sonnet, and asking it to explain the distinction between a function being smooth and analytic, one aspect that appears interesting is that the OpenAI forcing function is apparently constructed out of "bump functions", meaning that it is not a uniform force acting upon the flow but rather a highly engineered pattern of pokes, localized in time and space, which as another commenter in this thread notes sounds similar to Maxwell's Demon - another theoretical force, that neither tells us anything about Brownian motion nor the 2nd "law" of thermodynamics.
    So, we'll have to wait for mathematicians to continue to analyze the proof, and determine to what extent is does deliver on providing insights into Navier Stokes, and any potential improvement to it, that was the goal of setting it as a Millenium Prize in the first place.
    <a href="https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf" rel="nofollow">https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...
busssard · 1790075296 · focus · HN ↗
"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!"
1. rsfern · 1790078126 · focus · HN ↗
  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)
  1. busssard · 1790088710 · focus · HN ↗
    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.
  2. fmbb · 1790110433 · focus · HN ↗
    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.
    dgellow · 1790111102 · focus · HN ↗
    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
    busssard · 1790255575 · focus · HN ↗
    yeah but the problem they solved is not really useful. the article is also quite clear on that
2. guywithahat · 1790110728 · focus · HN ↗
  > "Guys, Guys calm! You did not produce any useful results!"
  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.
davidguda · 1790080534 · focus · HN ↗
Does it matter if it did? They got the headlines.
ChickeNES · 1790110000 · focus · HN ↗
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?
1. dgellow · 1790110733 · focus · HN ↗
  No, that not the issue. If you look at <a href="https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf" rel="nofollow">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.
  1. hatthew · 1790111204 · focus · HN ↗
    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.
    ChickeNES · 1790112529 · focus · HN ↗
    yes!
    dragonwriter · 1790112658 · focus · HN ↗
    Fefferman and the larger math community are not the same actor; a divergence between their concerns is not surprising.
    ChickeNES · 1790113988 · focus · HN ↗
    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.
    dragonwriter · 1790119930 · focus · HN ↗
    > But Clay chose Fefferman to represent "the larger math community"?
    Surprisingly enough, being chosen by another actor as proxy for some group doesn’t actually resolve the problem that an individual may not always be an accurate proxy for the concerns of the group (and especially for the same descriptive aggregate group a generation after the proxy acts on their behalf.)
    scheme271 · 1790120340 · focus · HN ↗
    I'm pretty sure Fefferman's formulation of the problem got a lot of vetting before the Clay Institute accepted it. As such, it probably does reflect the math community's understanding of the problem. The "problem" such as it is, is that it might be the least interesting case especially given the solution that was obtained.
    [deleted] · 1790113176 · focus · HN ↗
    [deleted]
    etdznots · 1790117641 · focus · HN ↗
    A solution to any of the four options made by an LLM armed with a theorem solver and millions of dollars of compute would still not be that interesting to the math community since it is not likely that any new insights oe questions can be derived from the result
  2. ChickeNES · 1790112521 · focus · HN ↗
    > 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.
    dgellow · 1790144956 · focus · HN ↗
    The clay institute doesn’t represent the broader math community
  3. aesthesia · 1790125805 · focus · HN ↗
    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.
2. dragonwriter · 1790112782 · focus · HN ↗
  Yes. And more humans—in this case OpenAI researchers—similarly failed in choosing how to direct the AI tools.
  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.
3. SpicyLemonZest · 1790113704 · focus · HN ↗
  [delayed]
4. alkyon · 1790115353 · focus · HN ↗
  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
  1. perching_aix · 1790117420 · focus · HN ↗
    [delayed]
IshKebab · 1790110013 · focus · HN ↗
The goalposts are moving so fast you can barely see them!
Next year's headline: But did Skynet kill all humans?
1. dgellow · 1790110309 · focus · HN ↗
  That has nothing with moving a goalpost, it’s about understanding the actual result and digging into the details
2. gr_norm · 1790110388 · focus · HN ↗
  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.
  1. IshKebab · 1790111868 · focus · HN ↗
    Please. The headline is clickbait, trying to imply that OpenAI somehow screwed up.
    gr_norm · 1790112415 · focus · HN ↗
    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.
    IshKebab · 1790145142 · focus · HN ↗
    I did read it. I was talking about the title.
  2. curt15 · 1790120788 · focus · HN ↗
    Or people with a vested interest in pushing the frontier labs' preferred narrative that they've achieved AGI...even while frontier labs continue to hire human "Account Associates", "Android Engineers", "Applied AI Engineers"...
ActorNightly · 1790110126 · focus · HN ↗
Betteridge's law of headlines
1. dgellow · 1790110938 · focus · HN ↗
  That would be wrong here…
olliepro · 1790110799 · focus · HN ↗
Classic example of moving the goal posts. “Exploiting a loophole” is how you solve many great problems in math.
1. dgellow · 1790110868 · focus · HN ↗
  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
  1. krackers · 1790112280 · focus · HN ↗
    >and isn’t the one the math community cares about
    Then why was it allowed as an option in the millennium prize statement?
    SpicyLemonZest · 1790114253 · focus · HN ↗
    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.
    oliculipolicula · 1790153776 · focus · HN ↗
    Today that point is moot because.. the Mill problems that are least likely to have been solved are most likely to have real world impact..
    I'll take on the biased hope that most of 100 solutions to be released will not have any applications for at least 30 years..
    In order to counter my own fear that the 9 big names will not be able to get openAI to act "more responsibly" (whatever that means)
    SpicyLemonZest · 1790188092 · focus · HN ↗
    [delayed]
    oliculipolicula · 1790215767 · focus · HN ↗
    The point of the committee is to benefit Ant.. (maybe only by striking fear in OpenAI)
    aleph_minus_one · 1790115231 · focus · HN ↗
    > Then why was it allowed as an option in the millennium prize statement?
    Because quite some mathematicians only realized after this OpenAI Navier-Stokes announcement that this allowed variant is much less interesting than they originally thought. :-)
    xigoi · 1790145987 · focus · HN ↗
    Bocause they expected it to be solved by human mathematicians using techniques that could provide insight into the harder version.
  2. signatoremo · 1790120744 · focus · HN ↗
    Who is this math community? Since when did they form the consensus that this option is not at all what they care about, before or after they knew about OpenAI’s solution?
    What about Tristan Buckmaster and Levent Alpöge, did they also attempt to solve the same challenge? Didn’t they know it wasn’t interesting?
    dgellow · 1790145175 · focus · HN ↗
    > What about Tristan Buckmaster and Levent Alpöge, did they also attempt to solve the same challenge? Didn’t they know it wasn’t interesting?
    Yes, they were working on what is considered a niche case, the option C from the millennium statement for Navier-Stokes. The article explains that clearly, you can just read it and get the details
    oliculipolicula · 1790155953 · focus · HN ↗
    I wish there were some better way to see, before the IPO, how openAI's actions correlate with their bottomline.
    (I'm proAI (for the masses, but green) BUT antiopenAI and a bit less antiAnthropic)
    dgellow · 1790169057 · focus · HN ↗
    Dario is sexily dangerous until you read his essays in full then realize it’s just the same LessWrong sci-fi ideas at the core, all based on absurdly simplified thought experiments designed to justify their priors. Those people are deeply unserious and do not value existing humans. They already made their mind decades ago rereading the fact we need a machine god and that they should be the ones building it.
    Just the idea of making a conscious digital being, to then get it to process excel files for its whole existence is such an immoral concept
jonlong · 1790110812 · focus · HN ↗
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)!
1. hn_throwaway_99 · 1790115891 · focus · HN ↗
  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.
2. HarHarVeryFunny · 1790116586 · focus · HN ↗
  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.
  1. hotdog1492 · 1790121450 · focus · HN ↗
    Reminds me of Maxwell's Demon.
3. p-e-w · 1790121042 · focus · HN ↗
  > 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.
  1. dmix · 1790124722 · focus · HN ↗
    There's a big market in journalism to take down the popular thing in the news. "Everyone is wrong" and here is my article where I wildly exaggerates some minor details to justify the headline which made you click on the article.
4. dist-epoch · 1790121729 · focus · HN ↗
  Or a better framing: Clay Institute chose the wrong Navier-Stokes problem for a Millennium Prize.
  1. glimshe · 1790130720 · focus · HN ↗
    Yet it had remained unsolved until OpenAI's effort...
  2. seanhunter · 1790144504 · focus · HN ↗
    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.
5. casey2 · 1790162761 · focus · HN ↗
  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.
  1. cyanydeez · 1790165308 · focus · HN ↗
    AI, the new minority!
yieldcrv · 1790112555 · focus · HN ↗
treating LLM's under a separate apartness ruleset isn't going to go well
yababa_y · 1790113293 · focus · HN ↗
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).
ex-aws-dude · 1790115459 · focus · HN ↗
I dunno this sounds like some insane goalpost moving
1. jrflo · 1790133161 · focus · HN ↗
  “Yes they solved one of the 7 most famous unsolved problems in math today but they only did the easiest version!”
  At the current rate (if they keep burning tokens on it, which maybe they won’t given the backlash) RH will be proven within a year and there will be some other thing that means it’s not actually that impressive…
emmelaich · 1790118109 · focus · HN ↗
<a href="https://archive.is/52HlJ" rel="nofollow">https://archive.is/52HlJ
jongjong · 1790121697 · focus · HN ↗
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, one of the first slides said "Computers will do what you say, not what you mean."
1. dist-epoch · 1790121808 · focus · HN ↗
  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.
  1. jongjong · 1790149868 · focus · HN ↗
    Sure, but you gotta compare frontier AI with frontier human!
cpard · 1790124509 · focus · HN ↗
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
1. irieir · 1790175371 · focus · HN ↗
  Cringe it’s not that deep.
  OAI should just get on with it and more importantly - produce more stuff that positively benefits the vast majority of the population.
  1. rafterydj · 1790191193 · focus · HN ↗
    That doesn't help foster meaningful discussion. It's very relevant to discuss what we understand.
kittikitti · 1790135955 · focus · HN ↗
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.
chmod775 · 1790143879 · focus · HN ↗
There might be more value in having AI 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 edges of mathematics.
Maybe something interesting will fall over because of that, who knows?
nikitamalyavin · 1790216185 · focus · HN ↗
What will happen if we'll build a real-life test if the proposed counterexample? The exactly same force, as defined, shape of the vortex, etc?
We should either see the effect or get to understanding of how to correct the model.
Note that the force is finite and doesn't "know" the vortex config. The singularity should be achieved in a finite time.

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