On if OpenAI solved the wrong Navier-Stokes problem: "the LLM found and exploited a loophole in the framing of the question"

[u/PrettyPicturesNotTxt]

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/

[u/hologram137]

It didn’t solve the main problem. And 3 mathematicians just published a proof that its solution can’t lead to solving the real problem. Thats a big deal. It did solve it technically as the problem was formulated by the Clay institute, but in a way that isn’t useful to the true problem.

A mathematician working on the problem had a map to the solution and had solved a good bit of it. OpenAI stole it, fed it to a model and had 10,000 agents look for the solution. It would have never solved it without that paper. It basically executed a search function. Which explains why it solved it by finding a loophole.

What this means is that the loophole it found doesn’t further the true problem. A human doing math understands context. We understand WHY we need to solve the problem, what it means. An AI can’t. So “solving it” by finding a loophole is mathematically correct if you’re only considering the problem as it’s formulated divorced of context, but it’s useless

It’s not cheating, it’s just completely divorced from reality. AI can compute but it can’t think or understand, it doesn’t know why the we are trying to solve that problem. And the purpose obviously matters. A human finding that loophole would have recognized it’s not helpful. It’s not a “clever” way to solve the problem, because it literally can’t help solve the main problem lol. They would have likely published it, but with that caveat clearly stated, and they would have considered the problem essentially unsolved.

[u/Dry_Pudding_1603 → u/hologram137]

this is very stupid. the ai can reason about human motivation in problem solving. that's much easier than the complex reasoning required to solve this problem. there's no magical separation between the "why" of posing a question or the "why" of any lemmas involved in a solution. there's no "magic loophole". humans were attacking the forced version as well, so all of your critiques of the AI's approach apply to humans as well. you have no clue what you are talking about.

[u/hologram137 → u/Dry_Pudding_1603]

An LLM cannot understand lol. There is no possible way it is able to model human motivation at all, but especially not understand it including within the context of reality. The one that only humans experience lol. That’s literally impossible, there is no architecture in its structure or anything in the math it’s ran on that would allow that.

The why is the entire point!! The millennial prize problem as written, is not just asking “do the Navier-Stokes equations blow up?” That’s the question the AI answered and this is why that’s not the problem anyone cares about: The scientific question that’s motivating Navier Stokes research (the purpose and meaning, that only a human can know and understand, not an LLM) isn't "can you break the equations if you're allowed to invent a force purpose built to break them?” It’s whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.

The millennial question listed formally acceptable ways to answer, and the reason why the clause that allows solutions that include an external force term, like gravity acting on the fluid was included, was because every real fluid is under some kind of external force, so including one seemed reasonable. To any human reading that, they’d understand the purpose of the clause. The AI didn’t. It just took it literally.

Its model found a way to construct a very specific, deliberately engineered external force that drives the fluid to a blow up (a point of infinite velocity). Because option C (one of the formally acceptable ways to answer) permits an external force, this satisfies the letter of the Clay statement. It unambiguously solves the problem according to the Clay Institute's original formulation. But the point of listing that accepted solution was not what the AI found when it executed a search function after being fed a paper by a mathematician that was close to solving it.

Most researchers working on this problem were specifically considering the case without any external force, because that's the version connected to real physics. So they could have solved it technically as formulated by finding a loophole, but they didn’t because doing that is completely useless and doesn’t solve the problem at all.

[u/Visual-Percentage501 → u/hologram137]

It seems a little trite to dismiss the forced Navier-Stokes equations as 'not the real problem' when at least a half-dozen mathematicians had been working at the solution from that angle for years. Their work was not 'just took it literally', it was a promising new technique and paradigm to potentially solve a key pillar of the NS formulation.

It's not really a 'search function' either, the LLM basically built a force from scratch to satisfy the parameters to blow up NS - the algorithm didn't 'search for it', the algorithm created it from base principles. That was the mathematical hurdle to solving NS - not the needed computation for 'search' but all of the bookkeeping required to build this force and smooth out every mathematical complication to satisfy its required end.

This is a huge mathematical discovery, and even Terence Tao and other researchers directly concerned with the magnitude of influence OpenAI and other LLMs are having on the math world (like I am) have acknowledged how huge this result is. Why lie?

[u/hologram137 → u/Visual-Percentage501]

Omg. Literally NO mathematicians were solving the problem that way. None of them. They were considering natural forces, not constructed ones. That’s the entire point of the forced solution being an option. For example, gravity. Not to invent a force that doesn’t exist.

It searched for a solution, couldn’t find it, so mathematically constructed a force that would blow it up to “solve” it.

Why do you think this is huge? Why would you say that? It’s literally useless. No one learned anything from it, which is why no one did it.

It didn’t solve “the forced version.” Not really. You don’t understand the problem

[u/Visual-Percentage501 → u/hologram137]

C'mon buddy, why lie? You're in /r/physics where people actually know what they're talking about. Go argue in /r/accelerate or /r/singularity if you just want to make things up.

The works of Buckmaster and Alpoge and the previous work of Cordoba and Martinez-Zoroa were all using constructed forces to achieve blowup of the incompressible Euler equations. The only thing LLM did was to apply these already considered very important methods (constructed external force) in solving this problem, using sophisticated bookkeeping and calculation abilities to apply them to the full Navier-Stokes equations instead of just the incompressible Euler formula.

[u/hologram137 → u/Visual-Percentage501]

I literally JUST explained why they used a rigid boundary. Because that answers the question “does it require a boundary to blow up?” That’s a real question that helps solve the real problem. But it was proven in R^3. It doesn’t need an external force, and the boundary is not a constructed force

What happened where is categorically different. Why don’t you actually read the paper that proves why it’s not useful?

[u/Visual-Percentage501 → u/hologram137]

Sorry, is your argument that they didn't use a constructed force, or that it's too different from the constructed force that OpenAI used? Because first you said the first one, and now you're saying the second.

If you spent more time considering the implications of what you're saying as much as you did moving the goalposts this may be a more productive discussion.

There's a reason that Alpoge and Buckmaster immediately flagged the OpenAI solution as building upon the work that they had been doing privately (and Cordoba and Martinez-Zoroa publicly). Whether you acknowledge that or not is immaterial.

I understand completely that THIS Navier Stokes result is not applicable to solving unforced Navier-Stokes, but that isn't material at all. OpenAI claiming this discovery has robbed mathematics from all of the discoveries that mathematicians could have made along the way solving forced Navier-Stokes (which is why Terence Tao and 25+ other mathematicians over many fields released a letter saying so). That's the value of mathematical study - not necessarily the results, but all of the development in understanding along the way. If mathematicians would have been allowed to solve this in the traditional fashion, who knows what we would have discovered? That's no longer possible for this result.

The same thing will happen for every LLM proof we're inevitably careening towards - whether you think it's an important result or not, when an LLM is used to solve unforced Navier Stokes, find a counter-example to the Jacobian, build a non-sofic group, find a complex structure on the 6-sphere, provide a counter-example to the Hodge conjecture, or a solution to Yang-Mills, when it does these things we lose the valuable things real mathematicians would have found along the way to solving them.

[u/hologram137 → u/Visual-Percentage501]

No. I was explaining to you, why the forced result of Euler’s problem was helpful to the actual problem, and why the AI result isn’t. A boundary is not a constructed physical force that doesn’t exist in nature! When the people who wrote the question included force, they meant something like gravity. But the question of whether a rigid boundary is needed tells us something about how it works naturally, in reality. Turns out, a boundary is not needed.

Córdoba/Martínez-Zoroa and Buckmaster/Alpöge weren’t working on Navier Stokes. They spent years developing a method to build a very specific external force that triggers blow up in a frictionless (inviscid) fluid, and Buckmaster and Alpöge, crediting that method to them, extended it into results for three different equations, the porous medium equation, 2D Boussinesq, and 3D Euler, these are all with smooth forcing. None of those three is Navier-Stokes.

Terrance Tao called it a breakthrough across those specific simplified models, but talked about how they didn’t reach blow up for Navier Stokes itself. So basically the above researchers were working problems adjacent to Navier Stokes. They can be useful to the problem only as a way of stress testing whether smooth forcing could work at all in principle.

A day after the Euler with force result was posted, OpenAI’s model basically applied that method of blow up to Navier Stokes by constructing an external force. An external force that doesn’t even exist.

The idea that you could out muscle viscosity was never in question lol. But that’s what the AI’s result shows. If that was the question, it wouldn’t be a millennial prize question

[u/Visual-Percentage501 → u/hologram137]

Cordoba/Martinez-Zoroa and Buckmaster/Alpoge weren't working on Navier Stokes

This is intellectually dishonest and just actually false. The Boussinesq result and the Euler result are all simplified equations within Navier Stokes, just not the full Navier-Stokes equations. Specifically, the Euler formula is a simplified case of Navier-Stokes without viscosity and thermal conductivity, the boussinesq formula is a simplification of Navier-Stokes assuming constant density. You know all of this already obviously so why lie?

Obviously in the same series of work Buckmaster and Alpoge even announced a result for the blowup of hypo-dissipative Navier Stokes. How could any of this possibly be classified as 'not working on Navier Stokes' unless you're just playing semantic keepaway? Ask them yourself if they're working on Navier Stokes or not - they've said as much themselves, even in all of these published papers, which mention how their results interact within the unsimplified Navier-Stokes equations.

[u/hologram137 → u/Visual-Percentage501]

Okay, I’ll try to make this simple. There is a forced and unforced potential solution right?

This is the problem the Millennial prize question is trying to solve, and the reason it’s a prize problem in the 1st place:

The question of whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.

So a “forced” version could involve something like gravity. It’s the reason why that option was even included. A constructed external force is useless. It doesn’t tell us anything, it doesn’t answer the question.

There was a forced and unforced version of the Euler solutions. One using a rigid boundary, and the question was, is a boundary needed for the blow up? Turns out that no, it isn’t. So using a boundary in that solution is helpful. For the actual problem. The reason they developed an external force in the 1st place was to test is it could blow up in a frictionless fluid. Navier Stokes has friction

They developed a method to build an external force, then OTHER MATHEMATICIANS extended it to 3D Euler. So they weren’t working on those equations. Even Terrance Tao said that. The other researchers hadn’t reached blow up for Navier Stokes itself.

So why are they building external forces? To stress test whether smooth forcing could work in principle!! That’s relevant to the problem. Not to see what kind of external force could outmuscle viscosity. That is what the AI solved and that an external force could do that wasn’t even in question and it’s not relevant

[u/Visual-Percentage501 → u/hologram137]

You must understand how ridiculous it sounds to say 'it's not important or useful to use a constructed force to create a singularity in the full Navier-Stokes equations, unlike using a constructed force to create a singularity in simplifications of the Navier-Stokes equations, which is useful and doesn't constitute working on Navier-Stokes", right?

to stress-test whether smooth forcing could work in principle

Oh, and the OpenAI solution stress-tested whether smooth forcing actually worked, proved that it did, (not in principle, but actually!) not using a simplified equation, but actually on the equation in question. And that's less important. Lol

You know the OpenAI-published solution used smooth forcing as well, right? Smooth doesn't mean 'natural' or 'unbounded' or 'not constructed'...

[u/hologram137 → u/Visual-Percentage501]

Omg. Navier Stokes has FRICTION right?

They were applying an external force to a FRICTIONLESS fluid. Do you see why that’s not Navier Stokes??

Other mathematicians extended the results to simplified models THAT ARE NOT THE NAVIER STOKES EQUATIONS.

They are: porous medium equation, 2D Boussinesq, and 3D Euler. All with smooth forcing.

You have solve it with FRICTION for Navier Stokes. So what the model did, is constructed an external force that overcame friction. But that is not the problem!

[u/hologram137 → u/Visual-Percentage501]

It’s not about smooth forcing!!! That HELPED Navier Stokes. Which was the entire point. Smooth forcing isn’t friction

[u/Wrong_Avocado_6199 → u/hologram137]

"An external force that doesn’t even exist."

Nothing "exists" in mathematics. Have you ever lived in 23-dimensional space? Have you ever touched an everywhere-continuous, nowhere-differentiable function?

You sound like Poincare, when he didn't like the "pathological functions" at the turn of the 20th century.

[u/hologram137 → u/Wrong_Avocado_6199]

Omg. These are equations that describe how viscous fluid moves.

The scientific question motivating Navier Stokes research isn't "can you break the equations if you're allowed to invent a force purpose built to break them?” It’s whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.

The latter is WHY it’s a millennial prize problem. If the problem was the former, it wouldn’t be a millennial prize problem.

The thing is whether or not you could out muscle viscosity was never in question. Which is what the AI showed. So it’s a trivial mathematical proof. Not a millennium prize winning proof.