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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#591

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.

With Lean, math has become a really well suited problem for LLMs. We will likely see large gains for many years from here, just doing more and more rlvr, like continuously, non stop. No need to train from scratch. It really doesn't speak to the general intelligence of models though. It does speak to how good these things can become when a problem space has verifiable rewards, especially when you can verify one step at a time like Lean enables.

Re: On the Navier–Stokes Millennium Prize Problem

#592

[flagged]

Even under your interpretation, OAI pushed a button and solved NS. Yes, that is very impressive. Are you kidding me? Imagine building an automated system that can solve NS.

Possibly after being given the significant part of the solution from actual human researchers. Which they then bullied. And they beat them to the finish line only because they heard rumor and threw everything at the problem. It doesn't look good for openAI in any way. I see more reasons to avoid using them rather than use them from this story.

Re: On the Navier–Stokes Millennium Prize Problem

#593

Earlier quoted context omitted.

Even under your interpretation, OAI pushed a button and solved NS. Yes, that is very impressive. Are you kidding me? Imagine building an automated system that can solve NS.

That's not my interpretation - that is literally what OpenAI say in that press release.

The "steal their thunder" is interpretation. What I'm saying is that you believe they solved NS on a lark to bully some other researchers, and that this is not impressive?

Re: On the Navier–Stokes Millennium Prize Problem

#594

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 exist…

Fields that allow verification, like math, will far surpass human level because they don't need human data for training. It's exactly the same as with Chess

Re: On the Navier–Stokes Millennium Prize Problem

#595

[flagged]

"Argh, they have rushed in too quickly to solve a Millennium Problem!" How far we have come :.)

There are bound to be a bunch more results like this, in math, physics, chemistry, and now that we essentially have a DeepBlue for math, a DeepBlue for physics, etc, these results are going to come.

SOME of the problems that have eluded humans are going to turn out to be low hanging fruit that are susceptible to this type of brute force (10,000 agents on a supercomputer running for 7*24 hours straight) AI search.

I'd be more impressed if OpenAI found their own problems to solve, rather than rushing in to re-solve one once they heard it was already solved (and therefore not so hard).

Re: On the Navier–Stokes Millennium Prize Problem

#596

Earlier quoted context omitted.

> I have a couple friends who did the Math tripos at Cambridge (so a pretty high level!) who work in tech and have unanimously said they have 0% expectations of an LLM doing a millennium problem anytime soon https://news.ycombinator.com/item?id=38433655 > Let's talk when we've got LLMs proving the Riemann Hypothesis (or any mathematical hypothesis) without any proofs in the training data. I'm confident in my belief t…

Will history look back at comments like these as people being dumb, or people trying to cope?

It's denial and coping. Most people i see show this tendency around AI which is also why it 's easy to be far ahead of most population nowadays

Re: On the Navier–Stokes Millennium Prize Problem

#597

Not a great time to be starting sophmore year in cs & math. Should I just say fuck it, and go hitchhiking across Europe with some friends?

Just don't. If you read the story here carefully, you see that AI was used to work from theory built by others which showed that the Euler equations possesed finite-time blow-ups. But to make that step, actual good understanding for mathematics was needed. My experience with software has been the exact same.

Re: On the Navier–Stokes Millennium Prize Problem

#598

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…

>If anyone has counterpoints to this I'd love to hear them! Sure. A proof without an unknown amount of human steering (and/or stolen research) would be an unquestionable achievement. To this day there's zero (0) evidence of any result by an LLM alone (maybe I'm wrong). If I just prompt ChatGPT right now with "give me a proof of the Riemann Hypothesis" and this thing delivers, I'm sold. But anything close to "yeah Cha…

The amount of goalpost moving is insane. "Yeah it can solve Millenium problems, but can it do it with nothing more than a one sentence prompt?"

Also there are proofs where the only human steering was "keep going".

Re: On the Navier–Stokes Millennium Prize Problem

#599
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…

If the model has access to the "anonymized" data from chats, and the model is capable of building its own context from data that it can search through, including this data. Then it looks pretty damning. An independent review of the data traces from CoT and tool use involved in producing the result should make it clear one way or the other. Seems like discovery in a civil lawsuit could be very productive.

Re: On the Navier–Stokes Millennium Prize Problem

#600

[flagged]

There's no evidence that Anthropic did it first.

Yes there is - that OpenAI PR says that it was Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU.
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