Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014)

68 pointsposted 13 hours ago
by gmays
(terrytao.wordpress.com)

41 Comments

v64

13 hours ago

This is going around due to rumors and baseless speculation on Twitter [1] right now that Anthropic has solved the Millennium problem related to Navier-Stokes [2]

[1] https://x.com/AndrewCurran_/status/2096062392442724805 for example

[2] https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...

strangescript

7 hours ago

This is a step beyond baseless predictions. Tao also had a "weird" "hypothetical" comment about LLMs solving complex proofs with impossible to human verify Lean.

throwaway81523

5 hours ago

There are theorems like that now, like de Grey's lower bound for the Hadwiger-Nelson (unit distance graph) problem. He used a SAT solver to check that a certain graph with 1581(?) vertices is not 4-colorable. There's no way for a human to check that.

Even simpler, imagine Anthropic announces Goldbach's conjecture is false and they have a billion digit counterexample. Anyone can download it (300MB compressed), but how do you check it?

Doron Zeilberger for decades has expected incomprehensible computer proofs to eventually take over mathematics.

hodgehog11

8 hours ago

Why would they send it out for "expert review"? Every time, they have just made the AI generate a Lean proof. In fact, it seems like the most plausible direction to NS is computationally assisted detection of a blowup solution, which has fantastic automatic validation.

levocardia

4 hours ago

Anthropic sent out its Fermat's Last Theorem result to an expert on formalizing Fermat's Last Theorem in Lean, for what that's worth.

lumost

7 hours ago

How do you know the lean is correct? You don’t bet the two trillion dollar company on “the ai said so”

Almondsetat

19 minutes ago

The surface of bugs in Lean is infinitely smaller than the human error involeved in a committee of peer reviewers. It's way more probable to say "it's proven because Lean says so" than "it's proven because a couple of reviewers said so".

Also, if a bug is found, all previosuly proven theorems can be reproven to immediately and conclusively find out if things went wrong somewhere

adrianN

6 hours ago

You carefully check that the problem is formalized correctly and then trust the Lean machinery to check the proof.

wiz21c

2 hours ago

Each word of your answer is carefully chosen. I'll add one sentence though: you let time do its job.

Of course there may be errors in lean, of course AI can take advantage of it, of course "carefully" is full of errors. So the only thing left is waiting to see if the result holds. And yes, it may take 30 years...

zarzavat

2 hours ago

As the recent "proof" of the Collatz conjecture shows, that's not enough in an adversarial context. Human mathematicians don't submit proofs that take advantage of soundness bugs in Lean. AIs do.

hodgehog11

5 hours ago

Exactly, and the advantage is that checking that the problem is "formalized" here is essentially isolated to verifying that the final theorem statement matches the claim. If there are no 'sorry's and the program compiles, then it has been proven. That's the point of Lean.

eru

3 hours ago

> You don’t bet the two trillion dollar company on “the ai said so”

Making an ill-advised press release hardly dooms the company. Just like the hugging face incident hasn't doomed OpenAI.

CSMastermind

12 hours ago

Lol as far as I know that post was the origin of that claim and it's clearly just a guy predicting something that will happen in the future with no information about it.

bee_rider

12 hours ago

For a second I thought they were aiming the scary proof machine at us mortals doing PDE stuff. Fortunately the speculation is just that they happen to be aiming it at a nearby mathematician type problem. Phew.

bilsbie

12 hours ago

If it is solved what are the applications of that? What changes?

margorczynski

12 hours ago

None really. It just says if the NS equations are realistic and can really model real physics or there exist some solutions that make it blow up (infinite energy). But even if that would exist (a solution that blows up) it doesn't mean it doesn't work for 99,999999% of the stuff we're interested in.

The question is basically a pure math question about PDEs.

lumost

7 hours ago

Maximally, a closed form solution would remove the need for Computational Fluid Dynamics. Any property could be derived from a (presumably expensive) analytic function.

Minimally. It could say that no such analytic function can ever exist. Which would be rather boring.

Turbulent fluids look awfully predictable with their spirals….

amluto

6 hours ago

That would be surprising IMO. We have closed form solutions to Newton’s Laws plus gravity (albeit not very many of them), we have several closed form solutions to Einstein’s equation in GR, and we have a whole lot of closed form solutions to Maxwell’s equations. But we still use numerical methods to solve interesting problems in all of these fields.

neutrinobro

4 hours ago

Honestly, for practical engineering purposes not that much. The Navier-Stokes equations are an approximation for a mathematically ideal in-compressible fluid. Even ignoring compressibility, physical fluids in the real world are not continuous fields since they are composed of discrete molecules. However, for that small class of problems where an analytic solution can be found, then it means you can be confident in the answer (it won't blow up to infinity), and that there are no other alternate solutions to the same problem.

bobmarleybiceps

10 hours ago

if there's anything that would convince that LLMS are one of the biggest innovations ever, it would be this :-D

goldenarm

13 hours ago

@dang please can we add a [2014] to the title ?

thomasahle

12 hours ago

Yes please. For a moment I thought Terrence Tao had scooped Anthropic.

tacomonstrous

13 hours ago

This is essentially irrelevant to the content of the post, but it's amusing to me that he casually mentions submitting to JAMS as if its acceptance were a mere formality.

gus_massa

12 hours ago

He has rejected papers too https://mathstodon.xyz/@tao/113721192051328193

Sharlin

11 hours ago

> With hindsight, some of my past rejections have become amusing. With a coauthor, I once almost solved a conjecture, establishing the result with an "epsilon loss" in a key parameter. We submitted to a highly reputable journal, but it was rejected on the grounds that it did not resolve the full conjecture. So we submitted elsewhere, and the paper was accepted.

> The following year, we managed to finally prove the full conjecture without the epsilon loss, and decided to try submitting to the highly reputable journal again. This time, the paper was rejected for only being an epsilon improvement over the previous literature!

hodgehog11

8 hours ago

It basically is a formality at this level. Many top math researchers now hardly even submit to journals at all and just put up a preprint.

At this scale, peer review happens by the audience. They don't need a journal to get people reviewing their work.

tacomonstrous

3 hours ago

None of this is true.

hodgehog11

an hour ago

Uh, care to explain? I have several colleagues that stopped submitting to journals once they reached full professor. They only submit papers from their students for the benefit of their careers. First-author papers, not so much.

immmmmm

12 hours ago

It’s pretty crazy what dynamics you get from NS.. until one realise they emerge from a tiny part of the solution space of Einstein equations.. which themselves emerge at the low energy limit of sth much bigger.

amluto

5 hours ago

Can you actually credibly find NS or even Euler’s equations as an effective theory from GR?

Euler’s equations and NS have this pesky velocity field, which requires the fluid’s state to be well described by a velocity at each point in space (and a density and a pressure, but I think GR has no problem with those). This means that you need some kind of interaction between particles to get them to exchange energy so that they thermalize instead of staying in the collisionless regime. (In other words, if you have two blobs of fluid collide, you need them to not go right through each other.) And I don’t think that GR is dissipating on the relevant scales.

As a real-world example, the universe contains neat structures (the horsehead nebula is a somewhat famous example) that are consistent with dark matter distributions that don’t really resemble fluids.

amelius

12 hours ago

Interesting to see that they are not using the coordinate-free representation (exterior calculus, differential forms) that mathematical physicists prefer to use today.

auntienomen

11 hours ago

Those notations are used when writing down the models, because they make clear the intrinsic geometry, the basic symmetries, etc. But they're not used so much in the study of solutions to the equations. Solutions tend to have peculiar features, tend to break underlying symmetries, etc. and there only needs to be one nasty particular solution to prove the NS conjecture wrong.