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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#661

Post by semi-extrinsic »

thangalin wrote: post #464
inkysigma wrote: post #270

Earlier quoted context omitted.

To be quite clear, the solution to the _Navier Stokes problem_ is one in which you get a finite time blow up (i.e. infinite pressure). This is more meant to suggest that Navier Stokes is unphysical in some way which is not necessarily unexpected. There's unlikely to be any engineering applications since even if the solution can be approximated, you still need to set up the initial conditions but at that point you can…

The proof of finite-time singularity may impact both fluid dynamics models (CFD) and AI reasoning models. Under specific conditions, Navier–Stokes equations allow velocity to grow infinitely, causing the continuum fluid assumption to break down. Knowing the exact mathematical breakdown mechanisms helps developers improve adaptive mesh refinement and sub-grid scale models around high-vorticity regions (like vortex str…

> Knowing the exact mathematical breakdown mechanisms helps developers improve adaptive mesh refinement and sub-grid scale models around high-vorticity regions (like vortex stretching and turbulent shear layers).

This is 100% wrong and reads like copy paste of AI slop.

Any simulation which uses sub-grid scale models is already solving a different PDE than the actual Navier-Stokes considered in the Millenium problem, and that PDE is guaranteed to have different properties. Full stop.

And to claim this is somehow connected to AMR methods is an example of the kind of pseudoscientific statement Wolfgang Pauli would have called "not even wrong".

Re: On the Navier–Stokes Millennium Prize Problem

#662

Post by semi-extrinsic »

fooker wrote: post #491
munificent wrote: post #217

Earlier quoted context omitted.

> This just pushes knowledge work further up the ladder, toward larger and more complex problems. You really think it makes sense for you to be higher on the "solving complex problems ladder" than the machines that solved fucking Navier-Stokes? I envy your self-confidence.

Yes, this is how science and engineering has worked for millennia. For example there are no engineering implications of this solution yet. For the next several decades, we'll have engineers (presumably with AI) optimize things like rocket engines and turbines and AC compressors to work a few percent better because the numerical approximations might have caused us to be overly conservative. AI is not going to magicall…

> For the next several decades, we'll have engineers (presumably with AI) optimize things like rocket engines and turbines and AC compressors to work a few percent better because the numerical approximations might have caused us to be overly conservative.

No. Just no.

Re: On the Navier–Stokes Millennium Prize Problem

#663

Post by closetheloopdev »

tristanj wrote: post #651
closetheloopdev wrote: post #396

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…

The entire drama is that OpenAI sniped a Millennium Prize Problem from an Anthropic-affiliated research team who had been working on the problem for nearly a year. In just 5 days. I don't think that can be understated.

I'm not here to judge since I don't have all the facts, but from what they announced: they tried all 6, found a probable lead to Navier-Stokes, concentrated efforts in that direction, and found a solution.

I hope the next solved Millennium Prize Problem will have less drama.

Re: On the Navier–Stokes Millennium Prize Problem

#664

Post by daveguy »

HDThoreaun wrote: post #601
20k wrote: post #311

Earlier quoted context omitted.

Yeah well, its easy to do if you steal someone elses work and then try to threaten them into staying quiet about it Edit: OpenAI have now admitted they were training on prompts at the time they made their breakthrough: https://mastodon.social/@tristanbuckmaster/11723647135247030...

All the ai labs are open about training on prompts. The question is if buckmaster had disabled that with the toggle openAI provides.

The question is more whether OpenAI can be trusted to honor that toggle switch. Given that they hold all chats for 30 days "for safety and security".

Re: On the Navier–Stokes Millennium Prize Problem

#665

Post by danielmarkbruce »

jrflo wrote: post #111
chinathrow wrote: post #82

Earlier quoted context omitted.

Pre-IPO marketing?

I'm so tired of this "It's just marketing!!" commentary. An AI model just proved one of the top 3 unsolved problems in mathematics, they have a Lean certificate showing it's valid. How much more evidence do you need that these models are actually highly capable?

Highly capable of writing math proofs, no doubt.

It's really unclear that this entire line of work (training LLMs for proof writing) has much real value outside of writing math proofs. It is reasonably clear that, similar to Deep Blue at the time, people are extrapolating the results to general intelligence because the people who usually write proofs are insanely smart (just like world class chess players).

Re: On the Navier–Stokes Millennium Prize Problem

#666

Post by vrganj »

bluebands wrote: post #366
pu_pe wrote: post #119

OpenAI thinks of this as a scoop, and it is, but the possibility that they trained the model on the prompts of the other mathematicians they were competing with will leave a terrible taste on every scientist's mouth. Seems like yet another advantage of using open models right here.

fwiw there is a big "TRAIN ON MY DATA" toggle you can turn off (that they almost certainly did) and Anthropic MTS are posting that they almost certainly did not "steal" their methods

The fact the toggle is on by default makes that only slightly less unappetizing.

Re: On the Navier–Stokes Millennium Prize Problem

#667

Post by zem »

20k wrote: post #234
ameliaquining wrote: post #220

Earlier quoted context omitted.

I don't know what anyone's been saying on Twitter and I don't care. If it's really true that there's a model out there that's that capable two weeks after the start of training, then that's objectively a much bigger deal than a priority dispute, even if the latter involves juicy allegations of espionage and skulduggery.

It isn't a priority dispute, the more concerning allegation is that OpenAI may be training their models on prompts that mathematicians were using to solve this problem, and then surprise surprise OpenAI were able to replicate that work in their latest model What we're really looking at is seemingly a massive plagiarism scandal, which especially brings a lot of the past results into question If OpenAI is training mode…

even apart from the plagiarism issue, what sort of slimy company thinks "oh, here's someone using our models to work on a problem, let's throw more compute at it and scoop them"?

Re: On the Navier–Stokes Millennium Prize Problem

#668

Post by fc417fc802 »

enraged_camel wrote: post #632
tedsanders wrote: post #237

Earlier quoted context omitted.

All of those statements sound true, based on what I've heard. - "very little human" input feels ambiguous, and if someone spends a few days prompting a model to solve a super hairy problem requiring a 100-page proof, I can understand reasonable people interpreting that as both "very little" and "not very little" human input - it's all true that a team worked on this, a bunch of compute was burned, and the problem was…

>> I'm not sure how any of this provides evidence that OpenAI took any of their work. Sorry, but the burden of proof lies in the other direction: OpenAI needs to definitively prove that their agents did not look at the existing work that was about to be published. Otherwise OpenAI simply stole the glory and the spotlight (and I'm being charitable here).

That's entirely unreasonable. Allegations of malfeasance always need to be backed up by evidence.

Re: On the Navier–Stokes Millennium Prize Problem

#669

Post by sobellian »

semi-extrinsic wrote: post #634
sobellian wrote: post #517

Earlier quoted context omitted.

NS is a question for natural science. Q: can we model these bodies of discrete particles with a continuous approximation? A: if you do, you can get aphysical singularities. "If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."

This is a wrong interpretation. Physicists have a shit-ton of models that produce "aphysical singularities", they just work around those to get meaningful answers anyway. This is a whole trope and stereotype. Some of the most successfull and accurate predictions in all of physics come out after you discard a bunch of singularities. See e.g. https://en.wikipedia.org/wiki/Renormalization Nobody who actually works in fl…

Whether or not ways exist to work around the singularities, that they exist is surely of note. Before von Neumann formalized QM people were still doing QM, okay fine. But it's wrong to then say von Neumann was doing no physics of note.

Re: On the Navier–Stokes Millennium Prize Problem

#670

Post by xyzsparetimexyz »

tiborsaas wrote: post #103
echelon wrote: post #53

Earlier quoted context omitted.

This is going to be dramatic in so many different ways. - First off, to reiterate, WOW. - Second of all, when does this end? Are we at the dawn of the singularity now? - People are saying OpenAI "stole" this from the work of an OpenAI user. If so, that's pretty fucked - how can we trust them? - Time to think about retiring from any knowledge work or business? This could be winner-take-all where a leading lab can butt…

2) We are witnessing the intelligence explosion from the first row, wherever this takes us 3) I'm still processing the drama, just found out about it after reading the blog post. If that happened based on private data, that's horrible. If that happened based on public tweets, then it's still abuse of power as OA employees access to compute (launching 10k agents) is quite heavy weight in boxing terms. But apart from A…

> But apart from AI and drama now that we have working solution to Navier-Stokes, what improvements can we expect in engineering?

Minor productivity boost in mathematics as people are no longer nerdsniped by the problem

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