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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#321

Post by kypro »

simianwords wrote: post #136
aizk wrote: post #52

People had joked a couple years ago "Well if they solve a Millenium problem it's AGI"... Well here we are.

> 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…

As someone with a background in AI and who has been playing around with neural nets for decades at this point, it's been genuinely amazing watching extremely intelligent people make confident predictions about AI capabilities and progress, then be so completely wrong.

There's a kind of theory of mind for AI (specifically neural nets) which I now realise I seem to have which is very hard to explain to people who haven't felt the magic of these algorithms. In fact, the algorithmic details almost doesn't matter at all. When you have a generalised learning algorithm really the only essential components are – compute, data and time. So long as you can scale these you can be certain you will also scale capabilities. There is never any exception.

That said, the capabilities neural networks tend to progress in step-functions rather than scale in correlation with compute, data and time, because algorithmic improvements tend to come every ~5 years and bring a significant step change in capability (or efficiency depending on what you measure).

I think people like Dario and others working at frontier labs see and understand this very clearly. And I suspect it's also why they worry about AI risk because even if you ignore the significant increases in compute and data these models are being trained with, it's concerning that it only took two real algorithmic improvements to take us from mostly useless predictive language models to AGI-level intelligence – and we're due another step change.

Re: On the Navier–Stokes Millennium Prize Problem

#322

Post by reducesuffering »

armchairhacker wrote: post #216
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…

Let's wait until AI solves a longstanding practical problem before "dawn of the singularity" (which could be tomorrow, but still).

Practical?! The goalposts will keep moving until morale improves (narrator: it doesn't)

Re: On the Navier–Stokes Millennium Prize Problem

#323

Post by numeri »

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…

That's such a shit parallel example that it borders on dishonest.

There are hundreds of incredibly strong scientific priors that would have to be disproven for the moon to contribute to the solution.

If a model was trained on this data, even if it was trained using methods that lead you to believe it unlikely to have learned details about the proof (e.g., maybe it was only used to train some kind of reward model, which played a minor role in the overall training and would thus be very unlikely to transfer details of a proof), you wouldn't have to disprove large swathes of known science to be wrong.

Re: On the Navier–Stokes Millennium Prize Problem

#324

Post by jaccola »

light_hue_1 wrote: post #38

The real story here: the priority dispute and its implications on AI. When your hosting provider has unlimited resources to throw at any problem, all they need to know are the good problems, and they can learn that from your logs, how can you trust them? They could easily have looked at the logs. We don't know. We'll never know! You can't trust places like OpenAI or Anthropic with your IP if you're a business. They c…

I think we can follow the incentives. We know…

Re: On the Navier–Stokes Millennium Prize Problem

#325

Post by igleria »

hkmaxpro wrote: post #304
colinhb wrote: post #122

Earlier quoted context omitted.

The allegations of contamination (using Tristan and Levent's work) aren't very well evidenced, but this behavior by OpenAI (from the authors' statement) makes them seem like the bad guys: > I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?” I replied that I am an academic, and asked why he thought going public would ruin…

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”.

If I was a company with a zero data retention contract involving OAI I would be asking for a third party audit of such claim of zero retention like, yesterday.

Re: On the Navier–Stokes Millennium Prize Problem

#326

Post by [deleted] »

mlsu wrote: post #302
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.

It seems like there were a couple of human mathematicians that were higher on the 'solving complex problems ladder' than this machine.

[deleted]

Re: On the Navier–Stokes Millennium Prize Problem

#327

Post by twobitshifter »

>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 use AI to solve mathematical challenges and claim prize money, not for scientific value alone or as a promotion of an AI lab’s capabilities)

My napkin math - If you get 33 output tok/s each agent will burn 10.5M tokens over 88 days. At $50/MTok (Astra cost), that is $525 per agent. With 10,000 agents, you’d spend $5,250,000 to get back a million.

(We also know that they were running more groups that varied in size and this model is a generation ahead of astra)

Re: On the Navier–Stokes Millennium Prize Problem

#328

Post by azan_ »

cherryteastain wrote: post #157
diehunde wrote: post #117

OMG this is going to affect the lives of so many people! We have definitively reached AGI

Navier Stokes existence and smoothness has approximately zero bearing on engineering applications

Existence of AI capable of solving millennium problem has enormous bearing on everything though.

Re: On the Navier–Stokes Millennium Prize Problem

#329

Post by rfgplk »

Aboutplants wrote: post #246
rfgplk wrote: post #107

Something I've been going on and on about for months now and no one seems to listen. LLMs today are allowing _anyone_ to access cross-discipline knowledge that was previously entirely inaccessible without a) extremely deep pockets or b) a massively talented and varied team. In fact, contrary to what the masses seem to think LLMs are actually _better_ at hard cutting edge physics/math problems than they are at fronten…

So, physical fields? I’m not catastrophic regarding jobs yet as I have an optimistic view of humanity in general and its ability to meaningfully survive, but the more time I spend thinking about the future of work, the more I’m leaning toward broad general abilities rather than distinct talents. To your point, I no longer need comprehensive knowledge of any particular subject, but what is absolutely valuable is “gene…

We are very likely at the begging of the next industrial revolution.

Re: On the Navier–Stokes Millennium Prize Problem

#330

Post by reducesuffering »

mlsu wrote: post #302
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.

It seems like there were a couple of human mathematicians that were higher on the 'solving complex problems ladder' than this machine.

Yes a couple of elite mathematicians working on the problem for a year, which AGI solved in a fraction of the time. What about everyone else 100IQ? What about as the models are even better 1 year from now, 2 years? The trajectory hasn't abated.
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