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The Navier–Stokes Millennium Prize Problem

simonwillison.net

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Re: The Navier–Stokes Millennium Prize Problem

#181

Post by jonathanstrange »

_bobm wrote: post #160

People are focused on the drama but the problem showing is the data. This is the elephant in the room and I am surprised that openai can be that stupid with it. How can openai do this, what is being claimed, at the scale of their entire userbase? If they do this only for particular sessions then how do they sieve through sessions for the good stuff? How are sessions stored, how are they processed, how much storage an…

It seems completely trivial to feed sessions to their own LLM and ask it to look for various things in them, from detecting problematic use cases to finding interesting mathematical work.

Re: The Navier–Stokes Millennium Prize Problem

#182

Post by Mawr »

aswegs8 wrote: post #157
sodic wrote: post #146

Earlier quoted context omitted.

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…

Reminds me of amateur table tennis. There is a strange dynamic where you down regulate your performance unconsciously when the opponent is playing worse and vice versa. Could be described as some physical form of this effect: https://en.wikipedia.org/wiki/Asch_conformity_experiments The term would be conformity / normative social influence.

Surely this has nothing to do with specifically table tennis, nor that it's amateur.

A better example would be mixed boys/girls sports classes in school, where the boys deliberately hold back as to not injure/scare the girls.

It's a pretty obvious and human thing not to go out and completely destroy a much weaker opponent. We're social animals after all.

There also may be an element of energy conservation, there's objectively no need to put in any more effort than necessary. Inefficient.

Re: The Navier–Stokes Millennium Prize Problem

#183

Post by pietz »

piker wrote: post #174
pietz wrote: post #171

Earlier quoted context omitted.

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."

That doesn't reject my claim. They just didn't name them in this post. It feels like, you're going through great lengths reading something into this.

Re: The Navier–Stokes Millennium Prize Problem

#184

Post by recursivecaveat »

aadyachinubhai wrote: post #17

LLMs can't contribute good code to some of the good OSS math libraries, How is it even solving these problems?

Notably all the major announcements so far are counterexamples or formalizations of existing results to my knowledge. Not necessarily something you can just brute force, but areas with high return on elbow grease.

Re: The Navier–Stokes Millennium Prize Problem

#185

Post by jonathanstrange »

sdcfgy wrote: post #29
junofan wrote: post #22

Earlier quoted context omitted.

Their privacy policy for normie subscribers says in plain English they use your Personal Data for research. I think it’s pretty unreasonable to use the service and expect otherwise.

If that is the case, why on earth would you use it in any professional setting?

I have the setting turned on in Gemini Pro even though I work on proprietary code because 1. the setting allows for some (very limited) "memory", and 2. I consider my source code almost public even when it's not open source because I don't work on programs that involve extremely high level of know how or proprietary algorithms. It's mostly CRUD that can be copied in a myriad of ways, whether people use my methods or other methods.

If Gemini can improve based on my code and sessions (maybe doubtful but who knows) and others can benefit from it, that would be a welcome side-effect.

Re: The Navier–Stokes Millennium Prize Problem

#186

Post by piker »

pietz wrote: post #183
piker wrote: post #174

Earlier quoted context omitted.

"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."

That doesn't reject my claim. They just didn't name them in this post. It feels like, you're going through great lengths reading something into this.

No, sorry I was reading it correctly but they addressed the issue in the post itself. They make it clear they were aiming at all 7 problems looking for the "two" that were rumored to be solved. But they didn't go hog on NS until they made progress. See sibling comments. I misread it the first time to be a claim that they "heard one of 7 human intractable problems are solved and spent 15 million on the right one".

Re: The Navier–Stokes Millennium Prize Problem

#187

Post by civvv »

bonplan23 wrote: post #179
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.

How have we got to the place we are today then? Someone must have made the first steps onto uncharted territory, otherwise we would be in a homogeneous state frozen in time.

I am not saying that LLM intelligence can not be the same, that they are uncapable of dicovering new fields/problems that they have no training on. I am just pointing out that historically it kind of "must" be true that humans are capalbe of this, but we have yet to see an LLM do something like this, something radically "new" in a sense. All of these breakthroughs appear to me (not a mathematician) to be more a case of "digging" through millions of existing attempts/work, patching it together into a result.

This would already make LLM's one of the greatest tool mankind has ever made, but it has yet to display what I would consider a necessity for human level intelligence, which is this ability to discover entirely "new" things.

Would an LLM, given enough time and only the currently available trainingdata with no further input from humans, be able to solve something that was discovered tomorrow?

For humans my answer would be: maybe, probably, because this has been done historically.

For LLM's I would not be comfortable in claiming that they could. I think they would not be any better at this than traditional computational bruteforce.

Re: The Navier–Stokes Millennium Prize Problem

#188

Post by nicce »

latent-person wrote: post #147
nicce wrote: post #135

Earlier quoted context omitted.

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...

That seems to be indeed true. I guess it gets validated quite soon.

Re: The Navier–Stokes Millennium Prize Problem

#189

Post by harhargange »

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…

Not the hope, but telling them that an answer does exist. This ofcourse means now we are going to go into an even more darker cave next time we are looking for some gold and pathbreaking discoveries will become even more rare. Add to that, the fear of people not winning against ai and having fewer rewards, then fewer people even enter those fields or attempt problems over the next generation. AI erodes skills not at individual but at civilisational level.

Re: The Navier–Stokes Millennium Prize Problem

#190

Post by kzrdude »

caughtinthought wrote: post #55
kzrdude wrote: post #21

Earlier quoted context omitted.

On the contrary, a level-headed summary that gathers information from all the different sources is necessary.

Sounds like a great use case for an LLM

It's got 'max' and 'ultra' but I don't see a setting for level-headed.
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