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The Navier–Stokes Millennium Prize Problem

simonwillison.net

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Re: The Navier–Stokes Millennium Prize Problem

#141

Post by consp »

jojva wrote: post #132
PowerElectronix wrote: post #116

Earlier quoted context omitted.

Funny how they did not solve any of the other problems, just the one where there was already solutions to the NS with some restrictions in their chats, and their solution seems to derive from those.

They explain in the article that they eventually redirected all their resources towards NS.

So instead of dictating research to unsolved, or largely unsolved, problems, we are now as "a society" directing compute power towards sniping research outcomes.

I thought this was an ebay thing for people with too much free money, but it seems a bit larger.

Re: The Navier–Stokes Millennium Prize Problem

#142

Post by sdcfgy »

hn993302 wrote: post #138
sdcfgy wrote: post #14

My take home from this entire drama is that one should not use LLM services for confidential or proprietary information as they all seem to be run by assholes. And you’re sending them everything you are doing. Would you send your lab notebook to an asshole? Hell no. I say that as a mathematician (on paper) who perhaps surprisingly doesn’t give a crap about the problem itself.

Or if you're going to trust one of them, maybe it shouldn't be OpenAI

Why would I trust any of them?

Re: The Navier–Stokes Millennium Prize Problem

#143

Post by spwa4 »

karmasimida wrote: post #100
npiano wrote: post #78

Earlier quoted context omitted.

Why is that simple or plausible? Why is simpler or more plausible than lifting an almost-finished solution from a researcher's account?

Because it is not finished. Their follow up claim is that OpenAI’s approach looks like another proof they had been working on the side, but hasn’t published yet

1) Because AI models are 1000x better about following a problem to its conclusion than coming up with a genuinely new idea.

2) Because if we accept the facts ChatGPT only came up with its "new" idea after being told exactly what the new idea was by a mathematician (OpenAI doesn't dispute this btw). And OpenAIs story comes down to the usual "We didn't look at it, trust me bro", which is made more hard to believe because OpenAI only started their efforts after receiving news of what the researcher was doing.

Oh and OpenAI emphasizes that part of the researcher's progress was made ... on OpenAI.

3) And, probably, the researchers were likely stopped by token limits, and that's the only reason they were slower than OpenAI themselves, which is very, very unfair.

4) OpenAI's story "smells" (like so many AI stories lately). Supposedly the company's team asked ChatGPT about solving millennium problems, and out of all millennium problems it just happens to pick the one where a solution can be found in its chat logs?

5) Yet again it would be in good taste for these AI companies to just give this to the researchers (no shortage of difficult unsolved math problems, so if AI can solve them all, just find another one). But instead, yet again they're fighting about it.

6) OpenAI admits they only went after this problem, with a team, no less, after finding out which researcher went after what problem, because of how they thought it would affect ChatGPT's PR. They are demonstrating, in other words, their willingness to destroy human researcher's reputation for PR wins.

That's getting close to big tobacco level morals right there.

7) If anyone wants to verify how much OpenAI cares about the truth, just ask on ChatGPT about the copyright lawsuit outcome and how it applies to OpenAI.

You'll get EXACTLY the sort of responses you get from Qwen about Tiananmen ("we didn't do it, you have no data, everyone's lying and if we did do it, it was perfectly reasonable because " style argument. Try it)

Re: The Navier–Stokes Millennium Prize Problem

#144

Post by lucfranken »

Outside of this discussion about what is fair and not. This is so highly interesting to think through.

The amount of millions available to do those kind of research cases is practically unlimited.

There are an unlimited amount of cases to work on.

What a huge development would this give to both humans and the world in general. Because in the end better understanding gives new options.

It's deeply interesting that those things now get a concrete economical price which seems to be viable to extrapolate. The enormous additional "production" of knowledge will inherently increase the speed of all pieces of research and development.

Taken into account that it's used wisely, the risks with a strong force are always huge as well.

Re: The Navier–Stokes Millennium Prize Problem

#145

Post by davesque »

keremk wrote: post #70

Occam's Razor says: "They heard this problem is solved or about to be solved amongst the rest of the other problems. They prioritized this and put substantial compute with their newest model and solved it." I know everyone loves juicy rumors, theories etc. but honestly that is the simplest and most plausible explanation given the state of AI improvement now. Obviously spending 15 million on a problem is not a slam du…

Isn't it also a simple idea that a model designed to recall relevant information from its training data, which is also known to have been trained on data from user transcripts, would, in fact, reproduce directly relevant work by leading experts in the field? Seems like Occam's razor would apply to that situation as well.

We know that LLMs are trained to recall relevant info. We know AI vendors are using user transcripts to train models. Two plus two equals four, right? I mean, an LLM that failed to recall the transcripts of those researchers would be a bad model.

Re: The Navier–Stokes Millennium Prize Problem

#146

Post by sodic »

bob1029 wrote: post #24

> ... we heard rumors that two Millennium Prize problems had been resolved. Inspired by these rumors ... I've observed this exact effect last week. I made a discovery regarding a stepwise performance improvement in a codebase. I shared the benchmark results with a peer and within 12 hours they replicated the same. We had both been looking for this for years. I think giving someone hope that an answer exists might as…

I have a similar story, but perhaps even stranger.

I work for a startup. We often bring a wooden arcade with us to conferences as a marketing gimmick.

The arcade runs a single side-scrolling video game. You're running from a monster and dodging obstacles. The goal is to survive as long as possible, and your result is measured in meters.

There are always a few competitive guys who spend the entire conference taking turns to play it. And every single time, the same thing happens.

Say the current high score is around 200m. Everybody fails somewhere around that number: 190m, 186m... Maybe someone manages 210m. And the high score moves up at a snail's pace.

Then, a new guy shows up and gets something like 500m on his third try. From their next turn on, everybody easily does 450 or more, even though they were struggling to get past 200 just one turn ago.

What makes it stranger is that the game is dead simple. It's not like the new guy discovered a move that unlocked this capability. And it wasn't a lack of motivation either - they'd all been playing for an hour already. They just started performing better after seeing it was possible. There has to be a name for this phenomenon.

Re: The Navier–Stokes Millennium Prize Problem

#147

Post by latent-person »

nicce wrote: post #135
latent-person wrote: post #124

Earlier quoted context omitted.

To claim something verified in Lean is wrong, you need to either argue that the theorem was stated incorrectly, or that there is a bug in Lean (assuming no `sorry` etc, which is checked by comparator). The number of lines needed to prove it is irrelevant (other than checking for a bug in Lean gets harder).

That is the point. Someone must verify that the Lean matches the actual theorem, precisely as it should be interpreted.

Which has nothing to do with the total number of lines, it's just the theorem statement you need to check. Here is what they showed, which is under 300 lines with comments https://github.com/openai/NavierStokesAndEuler/blob/main/Com...

Re: The Navier–Stokes Millennium Prize Problem

#148

Post by pietz »

piker wrote: post #139
pietz wrote: post #128

I find the claims from OpenAI somehow more relatable and reasonable. - They threw compute on a problem another team/company was rumored to have solved to see what their secret model could do. - The texts I read do make it seem like OpenAI wanted to talk and share credit generously. - Imagine working on a frontier math problem with someone at Anthropic and not only do you use Codex but also through a non-business acco…

It’s fishy though that they heard one of seven problems was about to be solved and threw perhaps 15 million bucks at the right one. [Edit: they said "two of": "On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved. .. we launched an effort ... on all open Millennium Prize problems".]

Why in the world is that fishy?

Isn't that exactly what almost everyone would do given that they wanted to see how capable their model is and the tense competition they have with Anthropic right now? Stealing impressive headlines from your competitor is pure gold.

Re: The Navier–Stokes Millennium Prize Problem

#149

Post by OtherShrezzing »

PowerElectronix wrote: post #116
keremk wrote: post #70

Occam's Razor says: "They heard this problem is solved or about to be solved amongst the rest of the other problems. They prioritized this and put substantial compute with their newest model and solved it." I know everyone loves juicy rumors, theories etc. but honestly that is the simplest and most plausible explanation given the state of AI improvement now. Obviously spending 15 million on a problem is not a slam du…

Funny how they did not solve any of the other problems, just the one where there was already solutions to the NS with some restrictions in their chats, and their solution seems to derive from those.

Considering how language models work, you'd expect two distinct conversations on the same mathematical problem to have enormous crossover.

Re: The Navier–Stokes Millennium Prize Problem

#150

Post by isaacfrond »

Blikkentrekker wrote: post #101
sdcfgy wrote: post #14

My take home from this entire drama is that one should not use LLM services for confidential or proprietary information as they all seem to be run by assholes. And you’re sending them everything you are doing. Would you send your lab notebook to an asshole? Hell no. I say that as a mathematician (on paper) who perhaps surprisingly doesn’t give a crap about the problem itself.

That they do it is just concerning to me in that it says that home-ran models just aren't good enough. Surely researchers like this have the processing power to run them at home, they just don't have the processing power to train models of comparable level. This is something I feared would happen and where open source would be left behind. Maybe they can do something with crowd-sourcing computational power from volun…

> Surely researchers like this have the processing power to run them at home,

Nobody has that power. Certainly not mathematicians.

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