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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#501

Post by Fordec »

mrbungie wrote: post #430
Fordec wrote: post #378

Earlier quoted context omitted.

If humans can figure out to optimize to circumvent bottlenecks, I have no doubt each new bottleneck will also get routed around, just now automated.

We are not in an everything-has-an-API world yet, and it'll for sure take some time to get there.

For sure. Anyone who thinks that we're in the end state of what progress can be made simply lacks imagination. This is all going to keep changing and iterating for the rest of our natural lives. The only constant is change.

Re: On the Navier–Stokes Millennium Prize Problem

#502

Post by markgall »

qbit42 wrote: post #236
hexomancer wrote: post #68

> On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved What's the other one?

I heard Hodge conjecture? Third-hand rumor though...

I assume the rumor is a counterexample? Where do I go to get wind of these rumors?

Re: On the Navier–Stokes Millennium Prize Problem

#503

Post by adverbly »

intenex wrote: post #440

I think this is clear evidence that AI models are now at the far frontier of mathematics innovation and discovery and exceed human limits. This specific problem having had a $1 million bounty on its head and still remaining unsolved for 26 years after the bounty was placed is pretty clear evidence that many of the world's best human mathematicians would have solved this problem if they could have, and none were able…

> will soon far surpass that of humans

To be fair, I think it's still an open question about how far it might surpass human capabilities.

I think it's clear that its speed of development will be significantly faster, but it's technically not proven that the frontier and problems don't themselves become increasingly difficult faster than any acceleration in intelligence past the point of human training, data and existing knowledge.

Should this be the case, we would see a rapid broadening of development, and a slow advance in the frontier in such a way that might surpass the collective capabilities of people, but not by very far.

Re: On the Navier–Stokes Millennium Prize Problem

#504

Post by monster_truck »

10xDev wrote: post #190
chilmers wrote: post #152

Earlier quoted context omitted.

The implication from their last couple of published articles[1][2] is that they think they’ve achieved “recursive self improvement”. [1] https://openai.com/index/research-acceleration-view-inside-o... [2] https://openai.com/index/an-alien-mind/

Compute will always be the bottleneck even if this were true.

Based on the leaps in local inference speed in the past month, which have been absurd, I'm p confident we're going to whiplash from compute constrained to storage constrained.

Bit apples to oranges, but it reminds me of all the fiber we installed in the late 90s, certain that per-strand capacity increases were years or decades out, only to get massively rugged

Re: On the Navier–Stokes Millennium Prize Problem

#505

Post by monster_truck »

mzhaase wrote: post #268
arctic-true wrote: post #35

Buried under the drama is the fact that OpenAI is claiming that an internal model they’ve been training for less than two weeks is more than twice as capable in mathematics as Astra, which was only made public a week ago. Even if this improvement is limited to mathematics, that is an astounding feat.

The singularity happening under trump? We could have had star trek, instead we're getting the combine.

pick up that can

Re: On the Navier–Stokes Millennium Prize Problem

#506

Post by hkmaxpro »

concinds wrote: post #468
hkmaxpro wrote: post #304

Earlier quoted context omitted.

Both Sam Altman and Sebastien Bubeck admitted they only want Buckmaster to be the lead author on a rewrite of the OpenAI proof. https://x.com/sama/status/2097385167002415140 https://x.com/SebastienBubeck/status/2097379411691516310 A wake up call for using OpenAI models. If you discover something with their model and you work for a competitor, they “felt it would be inappropriate” for you “to author OpenAI’s work”.

Their own claim is that they wanted Buckmaster without Alpöge to lead a rewrite of OpenAI's Navier-Stokes work, not of Alpöge-Buckmaster's Euler work. No one can know if that's correct without proof but I don't know how you're reading it so differently.

They want Buckmaster to dissociate with Alpöge in a follow-up rewrite of OpenAI's work. (They only publicly admit “Buckmaster as the lead author”, but judging from Buckmaster’s statement, it’s pretty clear that don’t want Alpöge at all.)

Just suggesting to a mathematician to dissociate with their collaborator for a follow-up work, because their collaborator “is inappropriate to author OpenAI’s work”, is completely against the norm of mathematical research. As charm137 puts it in a comment below:

> This is like a researcher from CMU saying to an NYU researcher that their collaborator, being from MIT, is a problem - this is as ridiculous as that!

Re: On the Navier–Stokes Millennium Prize Problem

#507

Post by nairboon »

tedsanders wrote: post #181
lambda wrote: post #130

Earlier quoted context omitted.

Why can't they rule it out? Is even OpenAI unable to track the provenance of all of their training data? This is one of the major problems with these enormous closed models, and even most open-weights models, which don't disclose their training process or training data. You can never be sure what went into its training. Did it come up with an idea originally, or is it just plagiarising its training data? Are there ma…

To truly prove some incidental usage data made no difference we'd have to (a) identify any of their de-identified data that came from their usage of ChatGPT, (b) train a bunch of expensive giant models, and (c) ask them all to solve the Navier-Stokes Millenium problem until hitting some level of statistical significance. It's just not feasible to run experiments like this to prove whether a piece of data has an effec…

I think there is a much easier way to prove that the ChatGPT usage of Tristan Buckmaster and Levent Alpöge (possibly also the ChatGPT usage of Córdoba and Martínez-Zoroa, if they use it) had no influence on OpenAI solving the Navier-Stokes problem.

If the internal OpenAI model is as capable as you claim (being able to solve a Millenium problem without using unpublished insights built on years of work from mathematicians), then it should be able to demonstrate this capability again.

How about OpenAI solves another Millenium problem within the next two weeks, that doesn't coincide with the parallel discovery/solution of other teams of mathematicians, using ChatGPT for preliminary proofs & write-ups.

Re: On the Navier–Stokes Millennium Prize Problem

#508

Post by irthomasthomas »

dorjoycb wrote: post #13

It seems like some other mathematicians (not affiliated with openAI) have also (or close to) done this. A statement was posted about the surrounding events by one of the them: https://cims.nyu.edu/%7Etristanb/statement.pdf Also Terrence Tao's post: https://mathstodon.xyz/@tao/117233528517340774

"When a further trained version of our internal model became available over the course of the effort, we updated our agents to that model."

woah, this gives some credit to the rumor that openai finetuned a model over the course of a few days for this task, and maybe trained on the Chatgpt/codex history of the authors, including drafts of this research.

Re: On the Navier–Stokes Millennium Prize Problem

#509

Post by irthomasthomas »

peri-cl wrote: post #58
dorjoycb wrote: post #13

It seems like some other mathematicians (not affiliated with openAI) have also (or close to) done this. A statement was posted about the surrounding events by one of the them: https://cims.nyu.edu/%7Etristanb/statement.pdf Also Terrence Tao's post: https://mathstodon.xyz/@tao/117233528517340774

Buckmaster: > "I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer." OpenAI (i.e. this OP): > "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helpe…

Doesn't that count as plagiarism?
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