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On the Navier–Stokes Millennium Prize Problem

openai.com

961–970 of 1001 posts

Re: On the Navier–Stokes Millennium Prize Problem

#961

"Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens." At a conservative estimate of GPT 6 Astra pricing, this would have cost upwards of 15 Million dollars for anyone using the OpenAI API! To me this is the one silver lining. Yes, they can solve millennium prize problems, but it still costs a fortune.

The Millenium Prize awards are going to need to be raised!

Re: On the Navier–Stokes Millennium Prize Problem

#962
>"The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.

Diagram of a swirling vortex illustrating inward spiral and axial stretching. A snapshot of local incompressible motion. Orange marks faster angular rotation; teal marks slower rotation. Circulating speed also depends on radius. The trajectories show inward spiraling and axial stretching."

Hmmm, isn't that interesting!

(Side note: Apparently we can't "compress" something, but apparently we can move more units of that thing over a specific space in a specific time... hmmm, I wonder what the difference between those two concepts could be...)

Main Observation: The image on OpenAI's web page above, looks sort of like the one for the Hopf Fibration:

https://en.wikipedia.org/wiki/Hopf_fibration

https://www.google.com/search?q=hopf+fibration&udm=2

Re: On the Navier–Stokes Millennium Prize Problem

#963
post #933

Earlier quoted context omitted.

Recursive self improvement of their upcoming IPO value maybe. They are fluffy PR pieces otherwise.

I tried to build a procedural 3d asset pipeline for a specific use case. Before the Opus upgrade in November it was basically no way of doing this. I gave up very quickly. After November i tried again, and no model could build me anything relevant. Now it just works. Took me an hour to progress to a point were i'm happy. Whatever they do, progress is still real, still way faster than I assumed The list of Ubuntus 240…

What would being prepared look like? A house in the woods with a store of food? Knowing how to pick door locks and set up a militia in the desert?

Re: On the Navier–Stokes Millennium Prize Problem

#964

Earlier quoted context omitted.

This will be remembered as one of the biggest milestones in AI progress. The drama around it will at best be a footnote, just like hardly anyone caring about the drama around Poincare conjecture today.

I agree. Nobody cares and will care about the drama, it is just marketing. This is the point where were definitely have reached AGI.

To the extent the LLM copied the plagiarized work, no. Like a number of the Erdos problem solutions turned out to be based on forgotten literature. So the drama matters even for the assessment of the mega agent work.

Re: On the Navier–Stokes Millennium Prize Problem

#965

Earlier quoted context omitted.

There are lots of startups creating labs that can be managed e2e by agents. That will connect reasoning to the physical world and dramatically speed up the plan, experiment, reflect loop beyond what humans currently do in science R&D.

Maybe. Maybe not. Look at AI drug design - it's not really speeding up the important part - drug trials. There isn't really a coherent plan to use AI for the most complex part of drug discovery at all.

I work in pharma and I've advocated for directly training models that predict drug trial outcomes. I don't think we have the necessary support (data + algorithms) to produce accurate models in this space. I suspect that predicting drug trial outcomes is roughly isomorphic to understanding human biology at a fundamental level (and accepting that our existing models of how drugs work are extremely limited).

Re: On the Navier–Stokes Millennium Prize Problem

#966

[flagged]

Even under your interpretation, OAI pushed a button and solved NS. Yes, that is very impressive. Are you kidding me? Imagine building an automated system that can solve NS.

> OAI pushed a button and solved NS

Not exactly...

From here:

https://cims.nyu.edu/~tristanb/statement.pdf

"I was shown a prompt and told the internal research model had simply been given the problem statement. Levent had been told by Sebastien “very little human input” had been used. This turned out not to be true. Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem".

Re: On the Navier–Stokes Millennium Prize Problem

#967

Earlier quoted context omitted.

But that lead happened to be the same approach Levent and Tristan had found... Meaning they would have found it first if OpenAI hadn't spent millions in compute on following their lead to its conclusion faster than them.

It's forced vs unforced Euler, so it's not exactly the same. Since OpenAI has access to their training data, they can probably scrub through the data to find out whether there have been any mentions of the similar approach, and whether it only comes from Tristan Buckmaster or if it is in the training data before that. They'll probably have to kick off another fleet of agents to scrub through the training data to answ…

> they can probably scrub through the data to find out whether there have been any mentions of the similar approach

They did check and it was in the trainingset but they could not confirm if it was the /cause/ of it, which is a bit of a cop out.

Re: On the Navier–Stokes Millennium Prize Problem

#968
post #863

created a simulation of the solution to describe what's happening and why it's important for engineers, climate modeling, etc : https://navier-stokes-singularity-simulator.netlify.app/ (updated so that it works better on mobile)

Is it really important for engineers, etc?

In reality you can't have things like infinite velocity, so if Navier-Stokes is predicting blowups, then isn't this a problem with Navier-Stokes not handling some edge cases, not a problem with reality that engineers need to be concerned about?

Re: On the Navier–Stokes Millennium Prize Problem

#969
post #929
post #841

Earlier quoted context omitted.

What makes you think they were coherent?

They managed to solve a problem that was beyond current human ability.

That was the net effect (assuming what they solved was the actual problem and not a loophole in the problem statement or a lean bug). My point is they need not all work coherently to do that -- for example, for all we know 3/4 of them went off the rails, their results were pruned, and the relevant results came from a random subset that happened to produce something useful.

Re: On the Navier–Stokes Millennium Prize Problem

#970
post #849

Earlier quoted context omitted.

How can you possible say this sort of thing in context of what looks like a millenium prize being solved. I swear there's nobody blinder than those who won't see.

Because it seems like most of the work may have been done by human mathematicians and cribbed by OpenAI at the last minute

The human mathematicians didn't solve the Navier-Stokes problem, they solved the Euler problem. And they were extensively using LLMs to drive the work, as described in the Buckmaster statement.

Any way you cut it, this is a major achievement for AI, besotted with human drama over whose prompt should be recognized by the history books.

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