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The Navier–Stokes Millennium Prize Problem

simonwillison.net

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Re: The Navier–Stokes Millennium Prize Problem

#171

Post by pietz »

piker wrote: post #153
pietz wrote: post #148

Earlier quoted context omitted.

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.

If they were willing to spend 105 million it, perhaps. But that would be surprising. It’s fishy because they spent something like 15 million on the right one.

Because that’s the one Anthropic was rumored to have solved.

Re: The Navier–Stokes Millennium Prize Problem

#172

Post by piker »

aswegs8 wrote: post #165
piker wrote: post #153

Earlier quoted context omitted.

If they were willing to spend 105 million it, perhaps. But that would be surprising. It’s fishy because they spent something like 15 million on the right one.

Again, that is the point. There is rumours that this one thing would be solvable, so they focus on this, spending 15m on one specific thing, instead of 105m on many. How is that fishy?

I understood the rumors (as described by OpenAI) to be that “one of” the prizes was solvable. If they heard NS specifically was solvable and aren’t saying that, they’re intentionally obfuscating that.

Re: The Navier–Stokes Millennium Prize Problem

#173

Post by suls »

sodic wrote: post #146
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 tu…

Reference-dependent effort ("bunching" around a target) [1] is probably the closest thing. The existing high score was acting as everyone's reference point. Anchoring [2] is the more general cognitive version.

[1] https://www.nber.org/system/files/working_papers/w20343/w203... [2] https://en.wikipedia.org/wiki/Anchoring_effect

Re: The Navier–Stokes Millennium Prize Problem

#174

Post by piker »

pietz wrote: post #171
piker wrote: post #153

Earlier quoted context omitted.

If they were willing to spend 105 million it, perhaps. But that would be surprising. It’s fishy because they spent something like 15 million on the right one.

Because that’s the one Anthropic was rumored to have solved.

"On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved. Inspired by these rumors and by the step change in performance of our internal model, we launched an effort to evaluate it on all open Millennium Prize problems and a few other high-impact problems."

Re: The Navier–Stokes Millennium Prize Problem

#175

Post by MrToadMan »

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".]

Didn't they say that they launched an effort to evaluate their model on all open Millennium Prize problems, and then narrowed down to Navier-Stokes as the most promising after seeing results on simplified versions of the problems?

Re: The Navier–Stokes Millennium Prize Problem

#176

Post by throw-qqqqq »

protoman3000 wrote: post #53

If they wanted to solve a Millenium prize problem so much, why did they not try to solve P-NP instead? It boggles the mind.

Are you kidding/trolling? The P=NP problem is FAR more fundamental, and if proven true, would basically be a proof that e.g. public key crypto can be broken (NOT a description of how to though).

Basically, it would be a proof that all the REALLY hard (combinatorial) problems out there, have a much simpler solution, if we were able to find it.

EDIT:

NS is used daily in engineering and gas/fluid modeling. We sort of “know it works”. The smoothness proof is “just” formalizing what practitioners assume is true (very coarsely said, no intention to diminish the result!)

It’s a bit like the Collatz function IMO, empirical evidence isn’t proof, but we’ve got a huge amount of evidence for the behavior we’re trying to prove.

I believe P vs NP is a different beast entirely. We don’t even know which way the answer should go.

Re: The Navier–Stokes Millennium Prize Problem

#177

Post by intended »

The more I see of frontier lab economics, the less it seems like the earlier predictions on how they will grow hold.

This imbroglio looks like a point in the journey for a firm that is realizing it is not going to make money selling shovels during a gold rush, and that it has more to gain from just… being vertically integrated across an industry.

Open models have definitely hampered the ability to sell tokens at a premium, so mass market adoption is impossible.

But then take someone like Jane Street, for example. They self report making $30bn leveraging LLMs. It’s a defensible assumption that they are making profit on it.

Perhaps it’s more profitable for OpenAI/Anthropic to build their own funds. They have the capital, compute, they can afford to recruit teams and buy any IP/Data required.

This isn’t a fully fleshed out argument, but it is the first time it feels like the winds are changing.

Re: The Navier–Stokes Millennium Prize Problem

#178

Post by piker »

MrToadMan wrote: post #175
piker wrote: post #139

Earlier quoted context omitted.

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".]

Didn't they say that they launched an effort to evaluate their model on all open Millennium Prize problems, and then narrowed down to Navier-Stokes as the most promising after seeing results on simplified versions of the problems?

Yes, you're right. It seems like they claim to have started broadly and narrowed it down based on some progress. That leads to different questions but does answer my initial "fishy" point.

Re: The Navier–Stokes Millennium Prize Problem

#179

Post by bonplan23 »

civvv wrote: post #15

LLM’s seem very good at solving mathematical problems of which there is an enormous amount of exisiting work/attempts in their training data. This is an amazing capability, but does not convince me that these models are «thinking» or «reasoning» in the way a human does. A human mathematician could in theory categorize/discover an entirely new field of mathematics tomorrow, based purely on their «human intelligence»,…

If an entirely new problem, within a new field of mathematics were to appear tomorrow, I highly doubt a human mathematician would be useful at all on their own.
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