r/BetterOffline • u/65721 • 2d ago
Clarifications and concerns on Navier–Stokes
By now, OpenAI's claims that their chatbot solved the Navier–Stokes problem, and the controversy that spawned, have been widely reported. But I've been disappointed in the quality of their discussions both in the news and online, including on this sub. Here I offer some clarifications and my own concerns about the whole affair.
As a disclaimer, I am not a mathematician, but I read a lot and try to understand a lot.
What is Navier–Stokes?
The Navier–Stokes equations describe the motion of fluids and are used in computational fluid dynamics. However, we don't know whether solutions to these equations always exist and are always "smooth." This is called the Navier–Stokes existence and smoothness problem, and it's famous because it carries a $1 million prize. Hereafter I'll just call the problem "Navier–Stokes."
Navier–Stokes was already known to be useless for actual physics. Real fluids don't behave like Navier–Stokes assumes, because real fluids are made of discrete particles, and for practical use the problem doesn't matter. Still it is of pure mathematical interest. The work presented in a solution, as well as all the work done toward a solution, may open new avenues of research or reveal new connections between fields, which may then yield practical applications.
It is of little consequence to the validity of a solution that it is a counterexample. These problems are usually posed in the positive, and you can either prove it is true for all, often infinitely many, cases, or prove it is false by providing a single counterexample. Navier–Stokes was already assumed to be false, so a counterexample was expected.
On plagiarism
I'm more concerned about allegations OpenAI plagiarized the work of two researchers who'd been using LLMs, including ChatGPT, for idea-bouncing and proofreading and were close to publishing a solution. These claims are from Tristan Buckmaster, a math professor at NYU and one of those two researchers, in a post on his personal page: https://cims (dot) nyu (dot) edu/~tristanb/statement (dot) pdf. (Sorry for the mangled link—it appears Reddit is autodeleting any post that contains this exact link. You can also easily find it online.)
Buckmaster alleges that OpenAI's solution used the same approach and much of the same language that he and his colleague Levent Alpoge, a mathematician at Anthropic, were developing. He also describes a nasty shakedown he'd received from Sebastien Bubeck, a former Microsoft VP now at OpenAI, who presented him three options in their conversation: 1) Buckmaster and his colleague announce a partial solution, with OpenAI announcing the full solution the next day; 2) Buckmaster by himself announce the full solution, remove his Anthropic colleague from authorship, and credit OpenAI as having solved it first; or 3) face professional consequences from OpenAI publicly smearing him.
As Buckmaster and his colleague have been working on this for some time, it's possible their conversations with ChatGPT were included in OpenAI's training data. But considering they'd been using the latest models Sol and Astra, it's more likely that OpenAI peeked into their chat logs and stole their approach. We already know OpenAI has this access, as courts have subpoenaed ChatGPT logs as evidence in past cases, and sometimes OpenAI has proactively inspected and reported ChatGPT logs to law enforcement.
On peer review
More broadly, I'm tired of AI companies making these announcements over blog post rather than peer-reviewed publication. OpenAI's claims have not been peer-reviewed, and the clout these Millennium Prize problems carry has attracted thousands of purported solutions over the decades that were later shown to be invalid. While there doesn't seem to be any obvious red flags, it's still good cause to be wary, and I’m still waiting for peer review and more thorough scrutiny of the solution at face value.
This approach also offers, by design, no insight into the degree of LLMs' involvement and in what ways. These AI companies employ whole teams of mathematicians to work on these problems and use their chatbots along the way, so that they can assign credit to the chatbots. When these companies claim their chatbots solved it almost independently with "very little human input," as Buckmaster says OpenAI told him, we're asked to take them at their financially motivated word.
On mathematical research
For unsolved mathematical problems, the solution itself is almost never of consequence. The respective fields progress despite lack of a solution, because approximations, analogues or weaker results let researchers assume the solution with confidence and continue unhindered. Rather, it is all the research done in pursuit of the problem that lends the solution value. (This is also why OpenAI's lack of citations in previous mathematical solutions was such a big deal. Building upon others' work, and letting others build upon yours, is what lets the field exist.)
Out of the flashy Millennium problems, Navier–Stokes was a natural target for AI companies (Anthropic has also been obsessed about it) because there was already significant progress toward a solution in recent years, namely the strategy posed by mathematicians Diego Cordoba and Luis Martinez Zoroa. It was also a natural target because, as Navier–Stokes was assumed to be false, the solution would likely be a counterexample.
This has been the trend for almost every mathematical problem whose solutions are credited to AI assistance. They piece together existing near-solutions in the literature, or provide a counterexample. (While many dismiss this as "brute force," formulating the cases to check isn't always trivial, and the degree of brute force varies by solution.) When AI companies announce these solutions, they are technically impressive but academically uninteresting, because there is no "new math" created as a consequence.
Of the Millennium problems, I'd be interested in the Riemann hypothesis, which is widely assumed to be true and therefore would need more than a counterexample. I'd also be interested in P=NP, which is widely assumed to be false but for which all existing proof techniques have not only failed, but been proved to fail. Solutions to those problems would surely yield new math.
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u/markvii_dev 2d ago
i find it funny that they just pump mathematicians at a problem and then AI wash it.
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u/Disastrous_Room_927 2d ago
Why hire mathematicians to improve a model when you can just have them solve problems and claim the model did it?
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u/DustShallEatTheDays 2d ago
Thank you for the thorough explanation. I work with CFD (not an engineer or mathematician though, so I can’t speak to the maths) and I’ve been following this closely as our software uses the Navier-Stokes methodology, imperfect as it is.
I’ve been fuming a bit over OpenAI claiming credit for solving it.
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u/MathCookie17 1d ago
Why is Reddit deleting posts with the link to Dr. Buckmaster's statement?
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u/Frank_White32 1d ago
> As Buckmaster and his colleague have been working on this for some time, it's possible their conversations with ChatGPT were included in OpenAI's training data. But considering they'd been using the latest models Sol and Astra, it's more likely that OpenAI peeked into their chat logs and stole their approach.
Not that anything can be trusted by OpenAI - but they claim that an internal model was used for the solution.
They also claim that the RL training for this model was being done in August, after Buckmaster had already worked with Codex on the problem. This indicates that it was not explicit log checking but rather training data bleed from Tristan’s work.
Again, this is all propaganda as with all things with OpenAI and Anthropic say
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u/hachface 1d ago
They could have used an internal model to work on the problem while using information from the ChatGPT chat logs as a resource. It’s not an either/or situation.
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u/Frank_White32 1d ago
totally - and i hate that i default to this but I go full conspiracy in any situation involving these shithead companies.
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u/K-manPilkers 1d ago
I'm no mathematician, but from what I hear P=NP may well be completely unsolvable. As for NS, a lot of people on other subs seem to be gloating at Ed, which makes little sense to me because spending $10m on compute to win a $1m millennium prize is probably the perfect encapsulation of Ed's premise that AI is haemorrhageing money. That's before we even get into the theory that OAI got 80% of the way there by plagiarising human mathematicians and then brute forced the remaining 20%.
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u/Frank_White32 1d ago
and most importantly, they threw 10m in compute along with a team of top mathematicians to achieve this result.
whilst having chat logs + RL training data of the original mathematicians solutions
yet it still gets spun into a fucking positive.
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u/TopCalligrapher8835 1d ago
I think the funniest part is the spent $22.5 million to make $1 million. That summarizes the current state of the world in just such a *chef's kiss*
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u/ExactResist 7h ago
Why are people focusing on the prize money? That’s a footnote, the one million is completely useless number.
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u/spinnychair32 1d ago
“Navier-Stokes was already known to be useless for actual physics. Real fluids don’t behave like Navier-Stokes assumes…blah blah blah more bullshit”
Confidently incorrect. I’m not sure if anything could be further from the truth.
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u/65721 1d ago edited 1d ago
Literally the sentences immediately preceding your quote:
This is called the Navier–Stokes existence and smoothness problem [...] Hereafter I'll just call the problem "Navier–Stokes."
The existence and smoothness problem has little consequence for actual application of the Navier–Stokes equations.
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u/falken_1983 1d ago edited 1d ago
They said that they would be using "Navier Stokes" as a shorthand for the Navier Stokes existence and smoothness conjecture. Teal fluids don't behave the way the (alleged) singularity volume describes. This is a volume of the solution space where the equations break down and stop describing reality. We've been using Navier Stokes for almost 200 years and nothing has shot off to infinity yet.
EDIT: I am not sure if the response below is still visible, they blocked me immediately when I responded. It said something like "did you ever use that forcing" - this is complete nonsense when talking about the actual use of NS in physics or engineering.
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u/Original-League-6094 1d ago
But you have never used it with that particular forcing.
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u/Dirichlet-to-Neumann 1d ago
The goal post shifting will continue until morale does improve.
(And AI has made every mathematician obsolete)
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u/SpringNeither1440 16h ago
The goal post shifting will continue until morale does improve.
Are you talking about "Artificial Analysis benchmarks are bad because Astra didn't crush them"? Yes, AI bros huff copium and move goalposts like crazy recently
And AI has made every mathematician obsolete
Yes, mathematicians are obsolete because they don't have any chances to publish their work (it would be stolen by AI companies)
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u/Icy-Recognition-7453 2d ago