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On the Navier–Stokes Millennium Prize Problem

openai.com

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Re: On the Navier–Stokes Millennium Prize Problem

#341

Post by stabbles »

nradov wrote: post #274
imbusy111 wrote: post #83

Earlier quoted context omitted.

I feel sorry for whoever has to read and understand the solution. It looks like the typical convoluted unreadable mess I see the models generate for software. It might be technically correct, but gaining insight from it is just intellectual hell.

There's an opportunity to build a Lean "optimizer" which automatically simplifies existing proofs.

Yeah, code golfing for lean would be amazing, especially if they can make the proof to Fourier's Last Theorem fit in the margin.

Extra credits if it is proven that the proof cannot be reduced any further.

Re: On the Navier–Stokes Millennium Prize Problem

#342

Post by red75prime »

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…

"Our work is so much harder than their work that AI now does" is a refrain of the AI story. In technical terms you concern can be stated as "AI needs to be much more sample-efficient to not be bottlenecked by the speed of doing experiments." People don't find out all the relevant phenomena present there by holy spirit, after all.

BTW, there's also a problem of asking interesting questions that AIs aren't yet good at.

No one has found any principled walls of AI development yet. And empirical results are quite telling. So, I guess, those problems will not stand for long.

Re: On the Navier–Stokes Millennium Prize Problem

#343

Post by Kotlopou »

matteoraso wrote: post #194

This is undeniably epochal, but I can't help but notice that this is yet another example of AI disproving rather than proving something. Is this just a coincidence, or does AI slightly struggle with proving theorems?[0] [0] Struggle relative to its ability to disprove, not struggle relative to people's ability to prove theorems.

There has been the proof of the cycle double cover conjecture: https://news.ycombinator.com/item?id=48863490

Re: On the Navier–Stokes Millennium Prize Problem

#344

Post by mrbungie »

denverllc wrote: post #298
mrbungie wrote: post #178

Earlier quoted context omitted.

Of course AI was involved, you'd expect most mathematicians and researchers to use AI nowadays. This drama is about AI achieving impressive outcomes with little to no human intervention, as that would be signalling AGI.

> This drama is about AI achieving impressive outcomes with little to no human intervention That's not at all what the drama is.

Of course, as any drama, it has been developing into a lot more but the main motivation for OpenAI has been about winning that battle.

Re: On the Navier–Stokes Millennium Prize Problem

#346

Post by efavdb »

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…

>> Keep in mind: natural science is different. It's not always a matter of computation.

Math is like this too. The big problems they've been solving have been identified as interesting only through lots of prior effort.

Re: On the Navier–Stokes Millennium Prize Problem

#347

Post by cherryteastain »

bhouston wrote: post #98

What happens to real fluid in this particular cases? If the singularity is in the physical space? Is this just a result of ignoring things like friction and energy dissipation via heat, etc?

Navier Stokes assumes the fluid is a continuum. The smallest scales that it effectively models [1] are larger than the mean free path of the molecules in the fluid, measured by the Knudsen number [2]. Whenever a phenomenon in the Navier Stokes equations happens in a scale on the order of or smaller than the mean free path, Navier Stokes effectively is unphysical. So, this is a phenomenon in the equation we use to model the fluid, not a physical phenomenon observed in a real fluid.

[1] https://en.wikipedia.org/wiki/Kolmogorov_microscales

[2] https://en.wikipedia.org/wiki/Knudsen_number

Re: On the Navier–Stokes Millennium Prize Problem

#348

Post by vatsachak »

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.

Brain has loops and parallel connections.

Loops and parallel connections make transformer go brrr

Re: On the Navier–Stokes Millennium Prize Problem

#349

Post by mrbungie »

scurnus wrote: post #286
mrbungie wrote: post #161

Earlier quoted context omitted.

They are highly capable, no doubt about that, but: 1) We don't really know how they arrived to this result except that they had a lead and that they threw millions of compute at the problem. The article is written in a way that makes you believe that it was just an agent loop with little human intervention, but without any evidence. 2) If the threats are to be believed, it is concerning how far they are willing to go…

1) The article is written in a way that states clearly they threw a lot of compute at the problem. In api cost millions of dollars. 2) Millenium Problems have been the goal every AI company wanted to achieve since their diffusion, all companies have thrown a lot of resource to solve these problems, as they are very famous and scientists spent a lot of time trying to solve them. The first company to solve it will rema…

> 1) The article is written in a way that states clearly they threw a lot of compute at the problem. In api cost millions of dollars.

Did I say otherwise?

> 2) Millenium Problems have been the goal every AI company wanted to achieve since their diffusion, all companies have thrown a lot of resource to solve these problems, as they are very famous and scientists spent a lot of time trying to solve them. The first company to solve it will remain in history, despite all of you finding excuses about it.

I know, but I don't know how that relates to my point, which is about the way they are doing it.

Re: On the Navier–Stokes Millennium Prize Problem

#350

Post by Squarex »

twobitshifter wrote: post #327

>The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents… The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. The Millenium Prize is $1M, what is the ROI? (Edit: since I was not clear, and confused some - I mean for a hypothetical of a third party paying commercial rates to u…

The ROI is probably billions of increased pre IPO valuation.
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