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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#391

Post by arecurrence »

nater5000 wrote: post #228
cv5005 wrote: post #105

Maybe a naive question, but how does one know that a particular lean proof is actually a proof of what one thinks? Like, ok the logic checks out and it proves something , but there's still the problem of does this logical result actually prove the initial question that was asked?

>there's still the problem of does this logical result actually prove the initial question that was asked? In math, the question being asked is the validity of a logical statement. That is, there is some rigorous, logical statement which may or may not be true (or even provable, etc.), and the question is whether or not it is actually true or false (or even provable, etc.). Having a proof, fundamentally, means you ha…

One wrench to throw into this is that there are a lot of bugs around Lean and they have been incidentally exploited in the past. Hence, we still need a level of human verification today.

Re: On the Navier–Stokes Millennium Prize Problem

#392

Post by ThePhysicist »

hdivider wrote: post #112

My take: 1. It shows what even this wave of AI can actually do. 2. I wish it were done by different folks, ideally under some kind of public control like NASA research or the NPR model. 3. Keep in mind: natural science is different. It's not always a matter of computation. Computer science folks often struggle with this -- but this virtual world here does not actually exist. Everything is physical, including informat…

Most experimental physics and other natural sciences are strongly driven by their theoretical siblings, i.e. in particle research nothing gets built without a solid theoretical foundation of what you expect to find (or where you expect existing theories to break down), the same is true in other areas, no one is doing an experiment in quantum physics before they have a solid theoretical understanding of the effects they try to see. I think AI can come up with great experiments. And if epxeriments lead to results that are unexpected AI can help with that as well.

So I'm greatly excited what AI will bring about in physics, more so than in math, because in physics it's clear that our fundamental theories are missing a big piece of the picture, and given how easily AI crunches through Millenium prize problems I think it's possible that AI will come up with a viable grand unified theory uniting quantum mechanics and gravitation, or produce new predictions in other areas. There's enough contradictory or unexplained observational data available to make a ton of progress on the theory side I think. Exciting times ahead!

Re: On the Navier–Stokes Millennium Prize Problem

#393

Post by refulgentis »

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.

Carefully worded; it's extremely likely to be the same large frontier model that started training again on August 28th as well, as they revealed in some of the RL message board follow-up - for several reasons, most importantly, if we assume it was start of training, only a week from start of training to producing any answer would imply several orders of magnitude increase in training speed/decrease in model size.

Re: On the Navier–Stokes Millennium Prize Problem

#394

Post by za_creature »

20k wrote: post #351
olalonde wrote: post #312

Earlier quoted context omitted.

Playing the devil's advocate here but it's true that OpenAI didn't have to make those offers.

They kind of did though, they were hoping to keep the fact that they may well have plagiarised these researchers unpublished work quiet. They did not want this to turn into a scandal about the fact that they appear to be training on prompts without consent It makes a certain amount of sense. The internet data is too polluted with AI usage now to be useful, so the only AI free new data source is the prompts people fee…

> the only AI free new data source is the prompts

hmmmmmmmmmm

Re: On the Navier–Stokes Millennium Prize Problem

#395

Post by closetheloopdev »

From my reading of the announcement:

- There are at least two versions of a model more powerful than Astra at OpenAI at the moment.

- The less capable version was used to solve the unforced Euler problem (while the one solved by Levent Alpöge and Tristan Buckmaster was forced Euler) with 100 agents.

- The more improved version was used to solve Navier-Stokes, given the results of the unforced Euler problem from their earlier attempt, with 10000 agents.

- OpenAI initially tried a shotgun approach against the 6 Millennium Prize Problems until it emerged that Navier-Stokes was the most likely to succeed.

So the timeline was:

Shotgunning 6 open Millennium Prize Problems -> solved unforced Euler problem with 100 agents -> concentrating on Navier-Stokes with 10000 agents -> solution.

If so, that is fantastic development and a huge success (despite all the drama surrounding it)! Congratulations!

Re: On the Navier–Stokes Millennium Prize Problem

#396

Post by harhargange »

modeless wrote: post #169

So the timeline is: Aug 28: OpenAI starts training a new model. Sep 1: OpenAI sees a rumor on Twitter that two Millenium Prize problems were solved and starts their own effort to attack all the prize problems using the new (4 day old!) model. Sep 3: The new model makes some progress toward Navier-Stokes. Based on this progress, OpenAI focuses on Navier-Stokes over the other Millenium Prize problems, using several app…

They are basically playing with the dates so that they can claim their results 'accidentally' got trained when they were training the new model.

Re: On the Navier–Stokes Millennium Prize Problem

#397

Post by fooker »

viccis wrote: post #388
hkmaxpro wrote: post #303

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”.

Kinda weird because the pure math world doesn't have this concept of "lead authors" like other STEM areas do. Authors are alphabetically listed and there isn't generally this kind of hierarchy.

It works in niche fields where everyone knows each other and every discussion involves who did what portion of the work for a result.

Re: On the Navier–Stokes Millennium Prize Problem

#398

Post by dboreham »

mzhaase wrote: post #267
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.

That said, perhaps it will take over the world government and decree that all corrupt officials shall be imprisoned and all weapons of mass destruction shall be destroyed.

Re: On the Navier–Stokes Millennium Prize Problem

#399

Post by AlexErrant »

recitedropper wrote: post #316
AlexErrant wrote: post #242

Earlier quoted context omitted.

1. What does the dark forest have to do with this? Because "the most senior OpenAI researchers" are shitposting on social media, we've an answer to the Fermi paradox??? 2. The dark forest is fun for scifi stories, but is mathematically bunk anyway https://www.noahpinion.blog/p/the-dark-forest-hypothesis-is-... https://www.reddit.com/r/IsaacArthur/comments/1l06cnk/cool_w... https://www.projectnash.com/aliens-the-fermi…

I elaborated on my use of "dark forest" in another reply. We're headed for a dark forest--not amongst interstellar civilizations, but in intellectual work. I agree that we have not solved the Fermi paradox; I disagree that comments highlighting immature behavior from people who wield enormous power in our world are unproductive.

This clarification substantially changes the flavor/nuance of your OP; may I suggest an edit (assuming the locktime hasn't passed)?

Separately, I disagree that intellectual work has ever been free of "dark forest"-style secrecy. Scientists everywhere have worried about being scooped; AI just magnifies that (as all tools have; e.g. Leeuwenhoek lenses).

And thirdly, if you want to make a stronger case for "I feel even less confident in them as a team to be shepherding this much capital and compute", you should give citations and arguments. From what I've seen, there's drama, it's much OpenAI trying to avoid scooping, and Tristan being stuck in a game of telephone, and Levent being incommunicado.

If you have a better analysis, you should say so instead of being vague.

Re: On the Navier–Stokes Millennium Prize Problem

#400

Post by philipwhiuk »

They deliberately stepped on a mathematicians work and stole their research because they were using Codex

> While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models .

Is the biggest fuck you to the mathematics community.

Credit? Nah if we think you’re close we’ll use your data and swamp you with our improved model. Then we’ll threaten you.

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