jonlong 1 day 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)!

  • hn_throwaway_99 23 hours 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.

  • HarHarVeryFunny 23 hours 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.

    • hotdog1492 22 hours ago

      Reminds me of Maxwell's Demon.

  • p-e-w 22 hours 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.

  • dist-epoch 22 hours ago

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

    • glimshe 19 hours ago

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

    • seanhunter 16 hours 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.

  • casey2 10 hours 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?

    • cyanydeez 10 hours ago

      AI, the new minority!

bmacho 1 day 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.

  • tomjakubowski 1 day 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

  • HarHarVeryFunny 1 day 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.

    • adriand 23 hours 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.

      • pennomi 7 hours ago

        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 5 hours ago

          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 assuming 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.

          https://arxiv.org/pdf/2609.20803

          When discussing the proof itself, 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.

          https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...

chmod775 16 hours 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?

ChickeNES 1 day 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?

  • dgellow 1 day 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 1 day 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 1 day ago

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

        • ChickeNES 1 day ago

          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 22 hours ago

            > 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 22 hours ago

          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.

      • etdznots 23 hours ago

        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

    • ChickeNES 1 day 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.

    • aesthesia 21 hours 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.

  • dragonwriter 1 day 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.

  • alkyon 1 day 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 23 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?

cpard 21 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

  • irieir 7 hours ago

    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.

    • rafterydj 3 hours ago

      That doesn't help foster meaningful discussion. It's very relevant to discuss what we understand.

yababa_y 1 day 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).

ex-aws-dude 1 day ago

I dunno this sounds like some insane goalpost moving

  • jrflo 19 hours ago

    “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…

kittikitti 18 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 22 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 22 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.

    • jongjong 14 hours ago

      Sure, but to know where the real limits are, you need to compare frontier AI with frontier human! That's where the bar is.

ActorNightly 1 day ago

Betteridge's law of headlines

  • dgellow 1 day ago

    That would be wrong here…

yieldcrv 1 day ago

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

olliepro 1 day ago

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

  • dgellow 1 day 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 1 day 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 1 day 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.

        • oliculipolicula 13 hours ago

          Today that point is moot because.. including this one, the Mill problems that are most likely to have been solved are least likely to have real world impact..

          Extrapolating on that, I'll take on the biased hope that almost none of the 100 solutions to be released will have any applications for at least 30 years.. (besides PR wins for AI companies)

          In order to counter the fear (my own lonely one) that the 9 big names will not be able to hold the execs to account or get openAI to act "more responsibly" (whatever that means).. in the form of direct hits to new subs or partnerships or funding

          I plead guilty to any accusations of (vicarious) sour grapes or sympathy for the weak

          Point to you, because.. the committee would make more sense if they also bring Ant to the table.. unfortunately nerds will be nerds, so perhaps, to you, and I'll reluctantly concede, mathematicians deserve to be serfs

          • SpicyLemonZest 3 hours ago

            I don't think Anthropic needs such a committee because they don't make a habit of hill-climbing other people's problems.

      • xigoi 15 hours ago

        Bocause they expected it to be solved by human mathematicians using techniques that could provide insight into the harder version.

    • signatoremo 22 hours ago

      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 15 hours ago

        > 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 12 hours ago

          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. For me it's all about the personalities.. the people in oAI are deeply uncool. not so in Ant. Dario is still dangerous, but I guess sexily dangerous lol

          https://www.linkedin.com/posts/corywarfield_aigenerated-humo... )

          • dgellow 9 hours ago

            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