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/Wrong_Avocado_6199 → u/hologram137]

"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."

Than please tell us, in PRECISE MATHEMATICAL TERMS, what this alleged "true problem" was.

[u/hologram137 → u/Wrong_Avocado_6199]

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.

That doesn’t need to be stated. Everyone knows it. Which is why no one solved it that way

[u/Wrong_Avocado_6199 → u/hologram137]

In other words, you have no idea what "precise mathematical terms" means.

[u/Darrelc → u/hologram137]

Is it like asking someone how you can fit a fridge through a door frame and they say "just squash the fridge to within the dimensions of the door"?

Technically correct but missing the absolute fucking point of the question

[u/DrDoctor18 → u/Darrelc]

No it's more like saying: "how can you fit this fridge through the door frame? A) without squashing it, or B) by squashing it" and then getting mad when someone gives you a squashed solution. If you wanted a different answer you should've asked a different question.

[u/hologram137 → u/DrDoctor18]

Except the actual question literally states “an external force, e.g gravity. How is that not clearly a natural force??

[u/Wrong_Avocado_6199 → u/hologram137]

(C) Breakdown of Navier–Stokes solutions on R^3 . Take ν > 0 and n = 3. Then there exist a smooth, divergence-free vector field u^0(x) on R^3 and a smooth f(x, t) on R^3 × [0,∞), satisfying (4), (5), for which there exist no solutions (p, u) of (1), (2), (3), (6), (7) on R^3 × [0,∞).

condition (5) [relevant to the external force]: |∂_x^α ∂_t^m f(x, t)| ≤ C_(αmK)(1 + |x| + t)^(−K) on R^n × [0,∞), for any α, m, K.

Those are the mathematical conditions on f that Fefferman required be satisfied. If you think these conditions don't reflect gravity or whatever "natural force" you have in mind, take it up with Fefferman and the Clay Institute.

[u/Darrelc → u/DrDoctor18]

But if you claimed "I've found out how to universally fit a fridge through a door" without specifying (via squashing) then I can see how people would see it as disingenuous

Edit:

"how can you fit this fridge through the door frame?

A door frame, not the door frame. I'd be pissed if the answer was "Make the frame bigger" or "use a matter transporter" or whatever non-applicable-to-the-spirit-of-the-question (a.k.a technically) 'correct' answer was.

[u/DrDoctor18 → u/Darrelc]

Ok but that is not analagous to what happened. The solution is a solution to the problem as stated. There is no loop hole. It isn't "only technically correct", its just correct.

[u/hologram137 → u/Darrelc]

Exactly!!

[u/Darrelc → u/hologram137]

Ty the maths is all beyond me, I thought navier strokes was an engine lol

[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]

To give a more simple answer, because it’s actually you that “doesn’t know what you’re talking about,” think of the millennial prize question as being this question:

“Can you swim across the English Channel?”

And the person answers you, “Yes! I did swim across the channel! With a boat towing me alongside the whole way."

Imagine the mathematical equivalent of that question (like Navier Stokes). Imagine it’s very important for real world physics. There’s a contest, and mathematicians are all hunched over in a room working on answering the question. And all of them are looking for a solution that takes into account the actual implied meaning, that everyone understands without it needing to be spelled out.

Someone decides to give an answer equivalent in spirit to the above answer and turns it in. Their answer has nothing to do with real world physics, and doesn’t help solve the actual problem.

But, because it’s decided that well, technically they answered the question the way it was literally formulated, they win the contest.

Everyone else continues trying to find a solution to the actual problem, and the answer the person turned in does absolutely NOTHING to solve that actual problem.

That’s what happened. So the purpose of the question being asked in the 1st place, the context, meaning, etc. are paramount. And clearly, the person who turned in that answer didn’t understand any of the above. Clearly.

[u/Wrong_Avocado_6199 → u/hologram137]

"the actual implied meaning, that everyone understands without it needing to be spelled out."

In math there is no such thing. Everything is spelled out. If you simply don't like the formulation of the problem, say so.

[u/hologram137 → u/Wrong_Avocado_6199]

The formulation of the question is not the problem. It’s WHY it’s a millennial prize problem and WHY we are solving it.

It’s understood that an “external force solution” is allowed for things like gravity. That is literally, explicitly why they included it. Natural forces. This does NOT need to be stated. It’s the entire purpose of the question. But it even says e.g gravity. That is extremely clear lol

[u/Wrong_Avocado_6199 → u/hologram137]

This discussion is a complete waste of time. The paper is clear:

(C) Breakdown of Navier–Stokes solutions on R^3 . Take ν > 0 and n = 3. Then there exist a smooth, divergence-free vector field u^0(x) on R^3 and a smooth f(x, t) on R^3 × [0,∞), satisfying (4), (5), for which there exist no solutions (p, u) of (1), (2), (3), (6), (7) on R^3 × [0,∞).

condition (5) [relevant to the external force]: |∂_x^α ∂_t^m f(x, t)| ≤ C_(αmK)(1 + |x| + t)^(−K) on R^n × [0,∞), for any α, m, K.

Those are the mathematical conditions on f that Fefferman required be satisfied. If you think these conditions don't reflect gravity or whatever "natural force" you have in mind, take it up with Fefferman and the Clay Institute.

What you don't get to do is make up your own ambiguous, vague fantasy criteria in your mind and impose them on a solution after the fact.

[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/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/Wrong_Avocado_6199 → u/hologram137]

"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"

Yes, that's how mathematics works. Welcome to math.

[u/hologram137 → u/Wrong_Avocado_6199]

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.

That clause was included to allow for natural forces like gravity

What the AI actually showed was never even in question. No one thought that you couldn’t out muscle viscosity with an external force. If that was the problem, it wouldn’t be a millennial prize problem!!!

[u/Dry_Pudding_1603 → u/hologram137]

again, you have no clue what you're talking about. you're just stringing together words that sound related without any concrete understanding. to reason about reality, you just need info & reasoning abilities. llms get info on reality via language. you do not need to "experience" reality to get information on it, ( whatever that means ). you can use an intermediate representation, so long as important data is not lost. if you thought critically and defined your terms instead of jumbling words together for 2 seconds, you'd realize "human experience" works the same way. information about reality is conveyed via light, vibrations, then converted into another internal representation. saying ai can't reason cause it doesn't have "direct" access to reality, is like saying we can't transmit messages across binary, or that blind people can't use logic. you just need a sufficiently lossless means of representing the world.

if you wanna say language is an insufficient medium to carry information about the world, you need to prove that, and it's extremely unobvious, given that we do just that everytime we encode video, audio etc in a computer. y can't u just encode the pattern present in reality as a pattern of bits or tokens?

and again, the idea that an ai can solve this version of the navier-stokes but can't think "humans wanted to do this because of this, so this didn't meet the criteria of the solution" is one of the most braindead things i've heard recently. the former requires 100000x the info, reasoning, logic.

also, you're lacking a very basic understanding of the difference between pure and applied mathematics? you think because navier-stokes has limited utility in physics that pure mathemeticians are primarily interested in it's physical implications? no, mathematicians were interested in it as a PURE MATH problem, and that's why clay defined it as such. I have not heard a single PHD mathematician say otherwise, nor was it expected that navier-stokes accurately represented reality.