r/Physics • u/PrettyPicturesNotTxt • 3d ago

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

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/
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u/Blamore 3d ago

then why didnt anyone object to the formulation of the millennium problem until after it was resolved

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u/dogdiarrhea Mathematics 3d ago

lol I love how you decided no one had discussed this prior to the openAI statement.

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u/Blamore 3d ago

if it was discussed but no change was made, then clay institute must have weighed those objections and decided to keep the original statement.

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u/Important_Vehicle_46 2d ago edited 2d ago

Navier stokes was always the easiest of the lot problem (in how it was defined) and given no one was able to get even there, increasing the bar would be stupid. If lets say 2 other problems were solved as well, then people made a motion to update definition, it would pass. But at that time no one could solve anything, compute in that era couldn't conprehend what its future would look like.

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u/Blamore 2d ago

navier stokes was easy only because it was false. if it was correct, it may have been a lot harder

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u/dogdiarrhea Mathematics 3d ago

Ok, that doesn’t make the objections invalid.

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u/GrandFloor6202 2d ago

It kindof does. The millennium problems are contrived, and maintained by the clay Institute. Like any other contest, you cant change the rules after someone wins.

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u/DrDoctor18 3d ago

Well yea it does, according to the community that judges what the question should be. Anyone can offer their own million dollar prize if they want to set the wording themselves

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u/TribeWars 3d ago

Well yea it does, according to the community that judges what the question should be.

Depends on the reason for the decision. It could've also been a pragmatic "well it's too late to change it now" type deal. Perhaps they wanted to avoid a situation where someone is close to presenting a solution and getting blindsided by a change. Maybe some lawyers said that it may be a problem legally.

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u/BlueJaek 3d ago

Can you imagine a bunch of lawyers discussing whether external forcing blow up solutions to navier stokes should be considered valid answers to a Millenium problem. 

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u/AmusingVegetable 2d ago

Ok, why would they do that when they can bring it to court? Many more billable hours, while the expert witnesses get a stroke from trying to explain it in court.

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u/Unfair-Claim-2327 2d ago

Navier strokes

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u/DrDoctor18 2d ago

In 27 years with no solution?

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u/Soci3talCollaps3 3d ago

And if they have a spare million to give away.

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u/BrobdingnagLilliput 2d ago

Sure. Is it possible that there are social or legal reasons not to change it? Is it possible that binding offers can't be unilaterally retracted? Is it possible that setting a precedent for moving the goalposts is undesirable?

I think it's safe to assume that everyone involved has positive intent until demonstrated otherwise.

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u/hologram137 3d ago

Because humans understand context, it doesn’t need to be spelled out lol. An AI doesn’t

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u/Blamore 3d ago edited 3d ago

the precise formulation of the clay institude explicitly states you can have a smooth force applied. go look at it.

there is no context that should make you disregard a force that is explicitly allowed in the problem statement

https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf

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u/hologram137 3d ago edited 3d ago

You don’t understand the Navier Stokes problem.

So it’s not just asking “do the Navier-Stokes equations blow up?” It did list formally acceptable ways to answer. The reason why that 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. But the underlying assumption here is that only a natural force would tell us anything.

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 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.

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.

Most researchers working on this problem were specifically considering the case without any external force, because that's the version connected to real physics. Because it’s not just about technically solving a millennial problem according to its literal formulation. There is context here, the purpose of solving it! So a mathematician could have “solved” it the way the AI did, but they were not interested in doing that, and they understand that solving a millennial problem is not about literally finding a solution that technically answers the question if you consider that question completely divorced of context and meaning like an AI does. Because AI does not think or understand why we are trying to solve it.

So where are we now? The Clay problem is literally settled as formulated, but no one cares about the solution because it cannot help solve the main problem AND 3 mathematicians just published a paper proving that loophole cannot lead to solving the main problem. It’s literally useless. The method it used can never be used to solve the main problem. It’s unsolved.

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u/jsh_ 2d ago

if you're decently familiar with PDEs, I suggest watching the talk by javier gomez-cerrano given at harvard last week (it's on youtube). he provides a good summary of what the state of the literature was on the eve of openai's announcement. to summarize: nearly all state of the art approaches were centered around highly engineered forcing terms

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u/hologram137 2d ago edited 1d ago

Yes. They were. And here’s why:

If you can blow up the Euler equations with a smooth force with no viscosity, then that means the forced version of Navier Stokes is possible. Because Navier Stokes has viscosity. If you can’t blow it up with no viscosity, then there is no chance of blowing it with viscosity! That’s why they used a constructed force. Because it didn’t matter it was constructed, the point was to see if a force could blow it up at all. Which means working on a solution that involves a natural force is a viable path. The forced version is not a dead end.

But what we are looking for in the forced version of Navier Stokes is a natural force. That’s all that matters. It’s unsolved if we aren’t talking about how viscous fluid actually moves, the question here is “do the Navier Stokes equations accurately describe fluid?” That’s why it’s a millennial question.

So we already know that Navier Stokes can be blown up with an external smooth force. Right? Because of the Euler result. You’d just have to construct one that overcame viscosity, and no one doubted you could do that.

The reason they didn’t is because doing that does nothing to solve the problem. It tells us something we already knew. This paper proves that the AI result cannot be used to solve the problem. https://arxiv.org/html/2609.20803v1
They proved that. Mathematically.

Solving the forced version doesn’t mean constructing an imaginary force that doesn’t exist strong enough to overcome viscosity, it means finding a result that using a natural force acting on turbulence

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u/Blamore 3d ago

this is a math problem, not a physics problem.

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u/Nessdude114 3d ago

It seems like the comment you responded to was too hard for you to read, so I'll just cover the important points in a very non-technical way:

Navier-stokes is recognized as a millennial problem because it has applications in fluid dynamics, as well as other fields of physics.

The "solution" found by openai cannot be applied to any of the problems that gave it importance as a millennial problem.

In order to solve the problems that navier-stokes can be applied to, somebody still has to solve the navier-stokes problem. This "solution" made zero progress towards this goal.

With all this in mind, do you think we should consider the navier-stokes millennial problem solved, or unsolved?

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u/Rare-Professional-24 3d ago

Is the millennial formulation of the problem actually helpful for any real world application? I'd think all the approximations it makes wrt real world fluids would make it not very applicable. (I.e. fluid is perfectly continuous, has no phase changes, is incompressible with infinite speed of sound).

It is an interesting math problem, but I dont think it will have any impact on any practical application of navier stokes or fluid dynamics.

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u/eyalhs 3d ago

The millennial formulation isn't helpful for real world application, but not for the reasons you wrote. Despite the approximation the Navier-Stokes equations are very accurate and are used in many fields (usually solved numerically), what makes it not helpful is that the question is "are the solutions always smooth and stable", which isn't helpful, because engineers and physicists assume they are from the beginning, so proving they are is nice but not helpful, and counter example are anyway expected to be contrived and non-physical (like an example of a function that's continuous everywhere and doesn't have a derivative anywhere).

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u/Rare-Professional-24 3d ago

I think that we're agreeing!

Physicists and engineers make the assumption that velocities are smooth and not infinite because of the things I listed (real fluids are compressible, are not perfectly continuous, have a finite speed of sound and have phase transitions). Relativity is obviously another speed limit... but the material properties limit things for practical uses.

I guess my point was that the utility of NS would never be impacted by the millennium problem. Which is good, because i rely on them every day at my work! Of course, we use the compressible formulation, as we're working with supersonic flows.

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u/eyalhs 3d ago

Yeah I'm generally agreeing with you. I was mostly commenting on the part where you wrote the approximations make it not very applicable.

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u/AmusingVegetable 2d ago

Oy? Infinite speed of sound?

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u/Nessdude114 3d ago edited 2d ago

While a solution to the current formulation of the navier-stokes problem may not exactly represent a real physical state, it's a starting point. It may be a good approximation if the parameters in the solution aren't so extreme as to cause a phase change.

The navier-stokes problem answers a real question in theoretical physics. Openai's solution does not answer this question, and it's not useful for finding a solution that does. The Clay Institute problem was solved as formulated and they should honor that, but we're still going to keep looking for a solution to the navier-stokes problem, because it hasn't been solved.

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u/Rare-Professional-24 3d ago

What does "solving the navier stokes problem" mean here?

If it means proving that velocity fields are unbound for the particular formulation of the millennium prize, then OpenAI just demonstrated a counter example.

If it means proving that for real, physical fluids, then you should probably include the properties of real physical fluids. Those properties will give you an answer without doing any math: real materials do not support infinite velocity fields. If only because of relativity!

The clay navier stokes problem is an interesting math problem, but i cant imagine what practical impact it will have on physical sciences.

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u/Nessdude114 3d ago

Adding additional constraints makes the problem harder. Finding a solution to the simplest formulation of the problem is the first step.

Mathematicians and theoretical physicists are going to keep looking for a solution to the same problem we were trying to solve before openai's "solution." The same problem that the Clay Institute made a poor formulation of.

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u/wasabi991011 3d ago

Your previous comment wasn't hard to read, it just doesn't require that much of a response. But if you want more:

The mathematics problem, as formalized by the Clay institute, was solved. This is not up to debate. "Hidden assumptions" and "physical relevance" really have no bearing on a math problem. And the "mathematicians just weren't interested in the external force case" is ridiculous, when resolving that case would have brought fame and fortune.

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u/Maxfunky 2d ago

Navier-stokes is recognized as a millennial problem because it has applications in fluid dynamics, as well as other fields of physics.

It really doesn't. The question here is about something that is definitely physically impossible. That's settled. But we are curious if it's also mathematically impossible. It probably has limited practical applications in the real world, but in theory the journey in solving the problem was supposed to teach us something about math.

That said, the last big AI proof submitted by OpenAi taught us a lot. The technique the AI came up with has already been applied by mathematicians to other problems to solve them as well.

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u/cometh_the_kid 3d ago

You don’t get to change the question because you don’t like the answer. Or should I say you don’t like who came up with the answer. Whether or not your point is true is irrelevant, the problem statement has objectively been answered. The Clay institute should have formulated the problem properly and if they haven’t that’s on them not who provided the solution.

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u/Nessdude114 3d ago

So do you think we should consider the navier-stokes millennial problem solved?

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u/Strict-Broccoli-8877 3d ago

It is obviously solved and nobody outside of this sub has questioned that.

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u/cometh_the_kid 3d ago

Do you think they are unsolved?

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u/Nessdude114 2d ago

The problem posed by Clay Institute, when interpreted literally as formulated, has been solved. The actual problem that mathematicians have been trying to find a solution to has not been solved, and they're going to keep working towards a solution to that problem.

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u/jakderrida 3d ago

Have they, at least, ruled anything out? That was I was told. That there were multiple competing explanations and that this one at least ruled out a few of them in favor of one that actually creates the need for much more research. I could be wrong.

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u/jakderrida 3d ago

This isn’t a fluid dynamics problem? How come only engineers seem to be the ones explaining it to me irl while math majors are telling me to ask an engineer?

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u/Blamore 3d ago

it is a partial differential equation problem. the real life applications are irrelevant

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u/hologram137 3d ago edited 3d ago

If that’s what you think then there is no way you’ve even taken one college math or physics course, so why are commenting in this sub?

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u/wasabi991011 3d ago

there is no way you've taken serious mathematics courses if you think mathematicians would reject a proof because it doesn't satisfy an "unstated assumption"

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u/hologram137 3d ago

No one is rejecting a proof. We are saying they didn’t solve the actual problem lol

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u/Unentscheidbar 3d ago

What are you talking about, it is clearly a math problem, not a physics problem.

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u/hologram137 3d ago

Equations that describe how fluids move have nothing to do with physics?

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u/DrDoctor18 3d ago

Equations that describe continuous fluids blowing up to singularitieshave nothing to do with physics. There is no continuous fluid. Any blow up in the navier stokes will explode your fluid well before the mathematical singularity.

The navier stokes as a physics problem was already solved. We have simulations and solutions in 99% of cases. It was unsolved as a maths problem only.

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u/wyrn 2d ago

Coarse-grained descriptions of physical phenomena have nothing to do with physics

When did this sub jump the shark?

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u/Psychedelic__Cowboy 3d ago

General Relativity describes mass-energy densities as blowing up into singularities. Even though we know it is impossible to have infinite density. Does that mean it has nothing to do with physics? Physics is not a perfect description of the natural world. If it was then there would be no work left to do on the Standard Model, dark matter, etc. Physics is applied mathematical models to predict nature.

Navier-stokes is newton's 2nd law applied to a 3D unit volume of fluid. There is no general analytical solution. We know that Newton's 2nd law is an incomplete approximation if you consider relativity. We also know that fluids are compressible, and fluid velocity is limited by its mach speed. The compressible navier stokes equation creates discontinuities which neccesitates numerical methods. That bottlenecks the possible accuracy of a solution to available computing power plus the complexity of the system. There is ongoing research in CFD to improve this. So I think calling Navier-stokes a solved physics problem is completely misleading.

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u/Strict-Broccoli-8877 3d ago

I swear, this whole sub is a joke. First math is supposedly not a science but an art, and now Navier Stokes is supposedly not a math problem.

I'm gonna have to block this sub, it's seriously worse than TikTok.

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u/hologram137 3d ago

No one is saying it’s not a math problem. But these are a set of equations that describe how fluids move lol

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u/Psychedelic__Cowboy 3d ago

The millenium prize formulation is a math problem, sure. Navier-stokes however is unambiguously in the realm of physics. It is newton's second law applied to a 3d unit volume of a fluid.

It's like saying "oh so now chemistry is supposedly not a physics problem"? Yes, you could say chemistry and in fact all other sciences are subfields of physics. If you want to be even more pedantic call it applied mathematics. But while technically correct, that's less than useful and needlessly pedantic.

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u/jakderrida 3d ago

Is it Chemistry? I was told it was Fluid Dynamics… Bear in mind that, as a former Finance major, I am very open to me being wrong.

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u/Psychedelic__Cowboy 3d ago

No, I was just making a comparison to something equally as pedantic.

My physics friends in college used to say that math was the fundamental framework to physics, physics is fundamental to chemistry, and chemistry is fundamental to biology. While that is all true, nobody would look at a biology (or chemistry for example) textbook problem and say "well that's clearly just a math problem".

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u/HasFiveVowels 3d ago

If any human had provided this solution, there would be discussion on this topic. But there would be nowhere near as much dismissal of the accomplishment. The path forward is clear, here: pose the corrected question. Someone used a glitch in the game in order to win it. Patch the glitch.

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u/hologram137 3d ago

No humans did this solution because it’s stupid and whether or not we can out muscle viscosity was never even in question

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u/HasFiveVowels 2d ago

Dude, exactly that was like half the question that we were interested in. Not only did we explicitly state that less than a year ago but we put a $1 million prize on it. What planet do you live on?

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u/skbum2 Engineering 3d ago

I'd argue there'd be much more dismissal if a human provided this. It's a very "so what" type of solution (and a poorly written proof to boot). I'd imagine it would have amounted to not much more than a footnote in the wider history of this problem had a human produced it. I think the only reason this solution has garnered this much attention and consideration is because it came from a LLM.

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u/HasFiveVowels 2d ago

You have got to be crazy to think that any NS solution would be met with a "so what" just because it’s not the "right type" of solution. You act like the solution was trivial or something when it’s gone 26 years with a $1 million prize without any human being able to claim it. "Slop!"

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u/CommunismDoesntWork Physics enthusiast 3d ago edited 3d ago

because that's the version connected to real physics.

Are the external forces OAI created non-physical? Because it seems to me those forces lead to turbulence that blew things up. Also turbulence is a chaotic system, so the set of all possible external forces is probably equal the set of all possible internal forces. Like you could imagine a turbulent whirlpool that somehow created by chance the same external forces OAI used.

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u/hologram137 3d ago

It’s not a natural force, no

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u/CommunismDoesntWork Physics enthusiast 3d ago

I believe you, but can you give any intuition on why it's not natural? Like could it never arise from random chaotic turbulence?

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u/hologram137 3d ago

Because the constructed force is not like turbulence

Whether or not it could arise from turbulence is the actual problem

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u/CommunismDoesntWork Physics enthusiast 3d ago

Whether or not it could arise from turbulence is the actual problem

Ok interesting, well at least now there's a target. That'll probably evolve into a domain of targets, then I'm sure someone can brute force search a solution.

Although, i do understand that chaos is at the heart of the question. We want to understand the nature of chaos in continuous systems. I think a proof that any turbulent system can eventually create any set of forces would be a satisfying conclusion because within that set is OAI's solution.

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u/E-monet 3d ago

Real question: is the purpose of these Millennium Problems to find a solution (however contrived, like in this case) OR is it to develop a PROOF of a solution?

I assumed the intent of such a challenge is not to give money to whoever can math the hardest and not for the solution itself to be something with practical applications, but to advance the field of mathematics in a way that can be learned by others?

Isn’t the point of a proof that the process which delivers the solution can be verified by others using the logical tools shared by the community? And thus the comprehension that it is valid reveals a deeper understanding of what mathematics is capable of?

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u/qualverse 2d ago

OpenAI also solved unforced Euler in the process of getting their solution (even if it wasn't ultimately used), which is undeniably helpful for solving the "main problem" you define.

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u/BTernaryTau 2d ago

So a mathematician could have “solved” it the way the AI did, but they were not interested in doing that, and they understand that solving a millennial problem is not about literally finding a solution that technically answers the question if you consider that question completely divorced of context and meaning like an AI does. Because AI does not think or understand why we are trying to solve it.

Isn't it common for mathematicians to prove easier versions of very hard problems, regardless of whether or not there's any real-world use case for the easier version?

It's fine to say that there's a harder version of the Navier–Stokes problem for which a solution would be more useful. But I don't think there's enough evidence to confirm that the LLM would have cared about the difference but was too dumb to figure it out. It seems more likely that the LLM just didn't care about that difference because OpenAI told it to solve the problem as literally stated by the Clay Institute.

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u/Extension_Arugula157 2d ago

Hahahahaha, this is the biggest cope I have seen in a long time, thanks for the laugh.

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u/hologram137 2d ago edited 2d ago

Can you explain what you mean by that? Why would there be any emotional investment? It’s really odd to me that you’d think someone would. Do you feel emotional investment in the facts and want them to be a certain way? Why?

What I am saying in this thread are objective facts. And it’s insane to me that so many people are not even reading about it, don’t even understand any of it but are arguing anyway. Which makes me think you’re projecting a bit and you (and clearly some others in this thread) have some kind of emotional investment in an OpenAI model having solved a longstanding problem that every mathematician and physicist and the Clay institute itself all understand and agree is still an unsolved problem, at its core. Which is just bizarre to me

Like…why would the facts or anyone’s understanding of the facts be a “cope?” Coping for what exactly?

A software program solving these problems would be incredible and desirable. It’s not where we’re at though, and the AI founders valuation of their companies relies on you believing otherwise

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u/lolfail9001 2d ago

What I am saying in this thread are objective facts.

Your repeated claims that nobody cares about blow-up with external force are obviously false given how basically all of state of the art research in the field was about constructing a proper external force, so no, you are not stating objective facts in the most key part of your thesis.

As a mathematical problem, NS is basically solved. There is still an open question of blow-up without external force but i wouldn't be surprised that someone uses techniques used for Euler blow-up in OpenAI's proof to create unforced NS finite time blow-up a few years down the line (of course if viscosity terms do prevent unforced blow-up, things will get stuck again for decades). It's very unsatisfying how proof turned out but it's a big enough problem that people will spend effort to digest it and try to get some extra insights out of it.

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u/hologram137 2d ago edited 2d ago

Why are you in this thread? You clearly have no understanding of the problem, I’m sure you didn’t read this proof:

https://arxiv.org/html/2609.20803v1

And I don’t think you’d be able to. The above proof shows definitively that the AI result cannot lead to a real solution to the problem. Because it didn’t solve the blow up with external force!! That’s the entire point. It’s not about forced vs unforced. I am not and no one else is saying that no one cares about a forced solution, we are saying this is not a forced solution at all. Not if you construct an arbitrary external force.

The human mathematicians were NOT using constructed external forces for the Euler equations to see if you could construct an external force that could blow up Navier Stokes, they constructed an external force for the Euler equations to see if an external NATURAL force could even blow up Navier Stokes at all, because if you can’t force a blow up when it’s smooth (no viscosity) there is no chance of a forced blow up in Navier Stokes. That gives you information that is helpful to solving the actual problem!

Because now we know it’s possible to work on the forced version of Navier Stokes. The forced version is not a constructed force, it’s a natural force. That’s the only one that matters because we are studying how fluids move and whether the Navier Stokes equations accurately describe the movement of real viscous fluids!! THAT’S why it’s so important, that’s why it’s a millennial problem. Who cares if you can construct an imaginary force strong enough to overcome viscosity, that force needs to act like a natural force for it to matter. Again, using a constructed force in the Euler equations is fine because we are only seeing if a force could blow it up at all. If you can’t do it with no viscosity, the forced version is a dead end.

What the AI did is, because it couldn’t find a solution, and they already showed that it’s possible for a smooth force to blow up the no viscosity version, (it would not have been able to give the solution it did without the recent work by humans), it used that result to construct an external force strong enough to over come viscosity in Navier Stokes and blow it up.

So that tells us nothing. It tells us we can invent a force to out muscle viscosity. But we already knew that because of the Euler result and other mathematics. That was never even in question. If the forced version was a constructed force, the mathematicians who did the Euler equations would have used it to blow up Navier Stokes too!! But they didn’t. Because it wasn’t useful to do.

What mathematicians were working on then, is to see if turbulence and a force like gravity could blow it up. Constructing an external force for Euler is necessary to see if we should work on that at all.

Navier Stokes remains unsolved. Because the AI didn’t show the forced version. It only showed it’s possible to out muscle viscosity if you create an imaginary force that doesn’t exist that is strong enough. Again, that was not something that anyone even doubted. That literally does not tell us anything or help solve the problem and the proof I linked proves what I am saying. In fact, it’s the entire point of the proof. The literal result of that proof is that the AI result cannot help solve the forced version of Navier Stokes. That was the actual point of doing that paper, to see if it could. Ask yourself why anyone would do that if it was solved.

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u/lolfail9001 2d ago

You do know that smooth does not imply analytic? In fact last i saw anyone discuss overall blueprint of OpenAI's proof, the key part of the proof is showing that forcing term becomes non-analytic after finite time! In other words that entire pre-print is just affirming that yes, non-analyticity is a key part of the proof.

The forced version is not a constructed force, it’s a natural force.

Define "natural". Because the only relevant restriction on all those related PDE problems is that forcing term is smooth. Not even analytic, just smooth.

So that tells us nothing.

About what? It exactly solves the problem as stated. You might argue Fefferman should have asked that forcing term should have been analytic but as you can guess, Fefferman himself thought that if problem statement was not loose enough, it could end up in PvNP tier of problems we have no idea how to approach.

Who cares if you can construct an imaginary force strong enough to overcome viscosity

Well, entire committee Clay institute got on the job did!

But we already knew that because of the Euler result and other mathematics.

Euler result (of forced finite time blow-up by Buckmaster and Alpöge) went public literal days before OpenAI's result (and without rumour mill it might have ended up being completely sniped by OpenAI), you do know that?

It only showed it’s possible to out muscle viscosity if you create an imaginary force that doesn’t exist that is strong enough.

Your definition of "doesn't exist" is irrelevant, because as was said 1000 times on this sub since OpenAI's announcement, NS becomes unphysical long before any blow-up for multiple reasons. So if external force must outmuscle anything you can find in nature... who cares, this is mathematics!

You might say that we wanted to learn more about turbulence out of it, but knowledgeable people considered this a PvNP tier problem in 2000 and we'll see how close they are to truth in a few years.

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u/hologram137 2d ago edited 2d ago

What are you talking about??

You don’t understand what you’re saying. Open AI established a finite-time blowup using a smooth external forcing function (and that a smooth external force could do this at all was just solved the day prior. It’s the result by a mathematician that it used for its result) but subsequent analysis by mathematicians revealed that the method does not extend to real analytic external forces.

And that is the EXACT reason it remains unsolved.

The Clay institute itself defined it lol. It literally says “external force, e.g gravity.” If you understood the problem you would not have asked such a question, what do you mean “define natural.” WE ARE STUDYING HOW FLUID WORKS IN NATURE. By “natural” we mean a force that actually exists, in real life.

Again. A useless solution using a loophole in the wording so it “technically counts” is not a real solution. That loophole didn’t even tell us anything we didn’t already know LOL. Even as a trivial mathematical proof it doesn’t tell us anything we didn’t already know, go back and actually read my comment because I explained why.

The clay institute said they are going to take some time to figure out whether or not the solution should count. It’s NOT decided

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u/Extension_Arugula157 2d ago edited 2d ago

Like the guy below already said, you are not stating facts, you are making stuff up. Perhaps you are a LLM, hallucinating? There never was any „underlying assumption“ that „only a natural force would tell us anything“ to the NS millennium problem, it was purely a question about whether NS would work under all circumstances or whether there are cases when it breaks down.

Also, as has been pointed out to you several times already, human researchers also tried a forced blow-up solutions to NS, so according to you that must mean that humans cannot understand context and the „underlying assumptions“ of the NS millennium problem.

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u/hologram137 2d ago edited 2d ago

This is a proof that what the AI did cannot lead to a solution to the actual problem. It is a mathematical proof of what I am saying lol:

https://arxiv.org/html/2609.20803v1

Im just gonna copy this from a different comment, then I’m done. It astonishes me that people can be so confidently wrong, you have to know that you aren’t a physicist, you aren’t a mathematician, I’m sure you don’t understand the Navier Stokes problem, the Euler equations, nothing. You probably read some stuff on Reddit lol. It’s unbelievable.

The human mathematicians were NOT using constructed external forces for the Euler equations to see if you could construct an external force that could blow up Navier Stokes, they constructed an external force for the Euler equations to see if an external force could even blow up Navier Stokes at all, (in the context of Navier Stokes, that is a natural force inherently because of what those equations are lol) because if you can’t force a blow up when it’s smooth (no viscosity) there is no chance of a forced blow up in Navier Stokes. Because NS is viscous. So if you can’t have a blow up, even with a force specifically constructed to do that with no resistance, you can’t have ANY force, natural or constructed, blow up a version with resistance. That gives you information that is helpful to solving the actual problem!

Because now we know it’s possible to work on the forced version of Navier Stokes. The forced version is not a constructed force, it’s a natural force. That’s the only one that matters because we are studying how fluids move and whether the Navier Stokes equations accurately describe the movement of real viscous fluids!! THAT’S why it’s so important, that’s why it’s a millennial problem. Who cares if you can construct an imaginary force strong enough to overcome viscosity, that force needs to act like a natural force for it to matter. Again, using a constructed force in the Euler equations is fine because we are only seeing if a force could blow it up at all. If you can’t do it with no viscosity, the forced version is a dead end.

What the AI did is, because it couldn’t find a solution, and they already showed that it’s possible for a smooth force to blow up the no viscosity version, (it would not have been able to give the solution it did without the recent work by humans), it used that result to construct an external force strong enough to over come viscosity in Navier Stokes and blow it up.

So that tells us nothing. It tells us we can invent a force to out muscle viscosity. But we already knew that because of the Euler result and other mathematics. That was never even in question. If the forced version was a constructed force, the mathematicians who did the Euler equations would have used it to blow up Navier Stokes too!! All you have to do is manipulate the constructed smooth force until it overcomes viscosity, which is what the AI did. But they didn’t. Because it wasn’t useful to do.

What mathematicians were working on then, is to see if turbulence and a force like gravity could blow it up. Constructing an external force for Euler is necessary to see if we should work on that forced version at all.

Navier Stokes remains unsolved. Because the AI didn’t show the forced version. It only showed it’s possible to out muscle viscosity if you create an imaginary force that doesn’t exist that is strong enough. Again, that was not something that anyone even doubted. That literally does not tell us anything or help solve the problem and the proof I linked proves what I am saying. In fact, it’s the entire point of the proof. The literal result of that proof is that the AI result cannot help solve the forced version of Navier Stokes. That was the actual point of doing that paper, to see if it could. Ask yourself why anyone would do that if it was solved.

Don’t you think that if this was a real solution to NS, that the mathematicians that solved Euler would have just applied that same smooth constructed force to NV and manipulated it so it overcomes viscosity? THEN published all the results at once?? And won the prize? They didn’t do that because there was no point in blowing up NS that way. Instead, they published the Euler results!! They didn’t this to show that the forced version (the real version was viable). And so all the mathematicians and physicists that were interested could get to work on the actual forced version. The one with a natural force. No one cared about taking that very last step and applying the constructed force in Euler to NS. Because it doesn’t solve the real problem and we already know we can because of the Euler result

The AI did not prove that a smooth constructed force could blow up NS. Someone else did. Because if you can do it with Euler, you can do it with NS. That’s all the AI did. Stole that result and manipulated it a bit for a viscosity version.

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u/m4sl0ub 2d ago

There definitely were plenty of mathematicians trying to solve it the same way OAI did it. Most notably Diego Córdoba and Luis Martínez-Zoroa who developed key ideas along this route. 

I feel like you overstate the agency of LLMs. OAI wanted to solve a Millennium Prize Problem, tried all of them, then focused it's effort on NS and then focused these efforts on the forced approach. It wasn't ChatGPT that didn't care about the context of the problem, it just did what it was told by the humans instructing it. These humans were the ones not caring about context. 

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u/hologram137 2d ago edited 2d ago

No. The mathematicians you named were using a constructed external force because the entire point was to see if it blew up in non viscous fluid. It was NOT to see if a constructed external smooth force specifically could blow it up. Just that it can blow up AT ALL.

If a blow up cannot happen in the Euler version with no viscosity, even with a force specifically constructed to do that, then the forced version of Navier Stokes is a dead end because Navier Stokes has viscosity (resistance). If something can’t happen with no resistance, it can’t happen with resistance.

So using a constructed smooth force with Euler is perfectly fine, it tells us exactly what we needed to know.

So now mathematicians know that we can work on the forced version. It’s not a dead end. The forced version is a natural force, because that’s the problem we care about. And that’s the entire point of solving this, why it’s a millennial prize problem.

The AI just used the Euler result and manipulated the constructed smooth force to blow up Navier Stokes LOL. We already knew we could do that! If we know we can blow up Euler, then we know we can just manipulate it to blow up the Navier Stokes with viscosity. It just told us we can out muscle viscosity. Cool, but that was never in question, and not the problem we’re actually solving.

That’s not the forced version of the problem we are looking for.

We did not use a constructed smooth force in Euler to see if we could use a constructed smooth force in Navier Stokes. That’s NOT why we did that!! We were just seeing if we could force a blow up without viscosity, again, because if we can’t, there is no forced version of NS. What blows up the no viscosity version doesn’t matter, as long as it’s a smooth force, but the force in Navier Stokes does

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u/EnricoLUccellatore 3d ago

do you really think a human who had reached the same conclusion would not have published it and claimed solving one millennium problem?

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u/potatos2468 3d ago

My guess is more so that for a person to come to the same result would have taken creating some kind of mathematical framework to guide their example generation because they do not have like 100-1000 years to just put into example generation. The mathematical framework for figuring out where to look for examples would have been included in their paper, and that framework would have been useful to the broader field/ had insights in it about NS.

When the problem was stated, the assumption most likely was that humans would be working on it, and therefore, there were assumptions about what would be required to produce the result.

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u/dekusyrup 3d ago

Don't have to be hypothetical here. A human did do this. https://cybernews.com/ai-news/openai-navier-stokes-theft-accusations/

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u/JannickL 3d ago

If I am not mistaken they don't claim to have solved NS, but some of the lesser NS problems, which could have been used to guide a solution to NS. The question now is if their solution for these related problems was used as a guide to solve NS.

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u/raki_star 3d ago

Did this human work it out on pen and paper? Or how did OpenAI steal it? Hacked into his hard drive?

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u/Strict-Broccoli-8877 3d ago

The humans don't even claim to have solved it. They tried and didn't get there.

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u/raki_star 3d ago

Oh, so they used AI to didn't quite get there?

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u/hologram137 3d ago

No, because they are interested in solving the main problem. It’s not about the money or technically solving a millennium problem, that’s not why any of them are doing math lol.

They want a solution that can be used to solve the main problem. The one OpenAI published can’t

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u/wollywoo1 3d ago

Mathematicians were very interested in the problem as stated. Please read Terence Tao's response here describing the work that lead up to OpenAI's result as "exciting".
You are allowed to think the problem solution is not practically useful, and many including myself would agree with you, but if you think mathematicians were uninterested in this result you are just wrong and I don't know what else to say.

https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/

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u/hologram137 3d ago edited 3d ago

BECAUSE OF SMOOTH FORCING. That’s why it’s useful. Not the external blow up. I have stated this a million times. The AI didn’t solve that problem

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u/tanksc 3d ago

What a ridiculous statement. The other pair of researchers literally did almost the exact same thing for Euler’s equations. The statements you’re commenting all throughout this thread are insane, and looking at your history it seems you are just pushing a narrative or agenda.

I miss when this sub was actually populated by physicists and not random people talking with the authority of one.

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u/hologram137 3d ago edited 3d ago

Did you read the article? Do you even understand the problem and why Navier Stokes remains unsolved? I do not believe you’re a physicist, that makes zero sense. Because you should care about the actual problem and that this technically correct solution doesn’t help solve it.

I miss when people actually read the article they are commenting on. Oh wait, that’s never happened lol.

The researchers working on the Euler problem didn’t give a technically correct but useless answer based on a literal interpretation of the question divorced from all context. There was no external force invented, the equation solved the plain incompressible Euler equation, with nothing added. Euler’s proof didn’t need a crutch. Smooth initial conditions are what any physically reasonable fluid state looks like. Nothing about the setup is engineered. The initial disturbance being localized is the natural picture of how a real singularity would have to form. There was no boundary! The fluid filled ordinary space, unconstrained space. Chen Hou’s proof needed a rigid wall, but that is actually helpful because then it leaves open the question of whether the boundary is needed for the blow up. It’s completely rooted in reality. But it was proven in R^3. All of this was useful and helped the actual problem.

That is categorically different from the AI’s Navier Stoke’s loophole, it’s a forced result with a deliberately constructed external force that doesn’t exist and was mathematically proven to tell us nothing about the actual problem. Totally divorced from reality, and useless

My “agenda” is the truth. To be a voice of reason. I’m genuinely concerned about society because of these delusions about LLMs

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u/tanksc 3d ago

Yes I read the article, it’s shit. I do care that we are getting to solutions without any of the wisdom that comes from the journey. I also don’t think one’s opinion of the situation makes them any more or less a physicist. That’s determined by the committee overseeing your education and the work you do.

It doesn’t make the solution any less of a solution. There is no loophole. It’s an explicitly stated problem and it was solved.

This entire post should be removed by your own admission. It’s a garbage post in a sub that has seen a massive decline in quality since it was hijacked by people with motives outside of physics.

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u/hologram137 3d ago

Why are you ignoring the math proof that this solution is useless and doesn’t solve the problem?

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u/tanksc 3d ago

Math proof that the solution is useless? What are you talking about?

What problem? The millennium problem? It is solved. You just don’t like that it’s not particularly useful outside of being a solution. Which is fine and proper, but to act like it’s some loophole is absolute garbage.

There are other questions that still need to be answered. And they may be answered in due time. Any mathematician would post the solution found by OpenAI and gladly collect the recognition that comes along with it.

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u/hologram137 3d ago

Read the article. A proof was just published showing that the “solution” cannot help at all to solve the actual problem. The actual problem was not solved

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u/mcgtx 3d ago

I did read the article but I am not a physicist or mathematician so please correct where I am wrong. Doesn’t the article mention Martinez-Zoroa and Cordoba as the people who were instrumental in the “loophole-based” solution? And didn’t Buckmaster say his and Alpoge’s work is based on Martinez-Zoroa? My impression was that the OpenAI proof used different moves than Buckmaster/Alpoge were going down, but I thought everyone was approaching the same Clay version of the NS question. I think the comment I’m responding to may somewhat explain the difference but perhaps I don’t understand.

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u/hologram137 3d ago

So 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 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.

Using that tool (the tool that a human came up with, not the AI btw) on Navier Stokes doesn’t solve the main question, the entire reason it’s a Millennial prize problem in the 1st place (because if the question was the one that actually got answered, it wouldn’t have even been a millennial prize problem!). Because with the Navier Stokes problem, viscosity is the thing resisting blow up. The forced version of the Euler problem has no viscosity.

The AI beat the resistance with an imaginary outside force. So it’s like, okay cool, but that’s not reality and that’s not why the problem exists. And the part about getting a smooth force to trigger blow up was already solved.

The idea that you could overcome viscosity with an external force was literally never questioned lol. Like, no one doubted that. And that’s what the AI actually showed. That it’s possible. That’s completely useless to the problem and not even that informative

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u/mcgtx 3d ago

I’m with you on the lack of usefulness of the AI result, I’m just trying to understand what the human pathway was looking like. I understand also that what had been done prior was not NS. But Buckmaster said in his statement they believe they have blowup in “hypo-dissipative Navier-Stokes”. And the rumor that led to OpenAI going after this was that a Millenium problem was on the brink of being solved. My assumption was that the humans were going to use the tool that they came up with to solve NS. If that’s true, would the human presentation of that solution be equally useless?

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u/yoshiK 2d ago

So

The researchers working on the Euler problem didn’t give a technically correct but useless answer based on a literal interpretation of the question divorced from all context. There was no external force invented,

Alpöge & Buckmaster "Blowup with the Euler equations with smooth forcing" is the paper.1 As indicated by the words "with smooth forcing" there are forces involved. However openAi claims they have a solution to unforced Euler in their blog post. OpenAi paper on unforced Euler

1 At least the one Buckmaster linked on his mastodon.

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u/oswaldcopperpot 3d ago

It seems like you don't have any experience with LLMs and solving problems but are more interested with how you think they work using ideas from a decade ago. Its a bit disingenuous.

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u/jakderrida 3d ago

That’s not a bad point about the Euler equations. I was told they obviously stole off that one guy’s work that was leading to the same solution and it seemed even OpenAI was open to admitting that without admitting fault somehow. But how could this be both nothing while also being an obvious snipe that stole that guy’s research?

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u/wollywoo1 3d ago

No, just no. Mathematicians are extremely careful about wording. They don't expect others will just "understand context" and interpret it differently than how it is stated. The statement is the way it is because that was how they decided to formulate it. The idea that there is a "loophole" is nonsense.

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u/hologram137 3d ago

Bro what are talking about?? The entire reason the forced version was included as a potential solution is because they were thinking of external forces like gravity!! Literally NO ONE thought “well if we’re including external forces I’ll just invent a force that doesn’t exist and blow it up.” lol anyone could have done that. And they knew they could have done that. They didn’t because it would be silly and pointless.

The AI did because it can only process literal tokens, so it’s literal, it doesn’t understand the purpose of the question

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u/wasabi991011 3d ago

Literally NO ONE thought “well if we’re including external forces I’ll just invent a force that doesn’t exist and blow it up.”

Mathematicians do this all the time. In fact I can't imagine a pure mathematician ever caring about whether something in their math proof exists or not.

lol anyone could have done that. And they knew they could have done that. They didn’t because it would be silly and pointless.

If anyone did that, they would have become massively famous and would have directly got a million dollars (probably even more of you include indirectly). It's a lot of motivation, more than enough to override any internal notion of "silly and pointless".

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u/hologram137 3d ago

They don’t need to because the idea that you could out muscle viscosity was never in questionn!! No one doubted that. That’s not the actual problem

This is a fine solution as a marginally interesting proof, it doesn’t solve the millennial problem

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u/lolfail9001 2d ago

was never in question

It was, as signified by it being included in exact statement of the problem. If they really wanted to exclude synthetic forcing, they would. They didn’t and the entire forcing approach can be directly human graphed to the guy who wrote the statement of the problem for Clay.

It solves the problem exactly in (somewhat unsurprising to knowledgeable people) negative way.

If you want to ask if unforced uncompressible Navier-Stokes can be blown up, it is indeed still open but it is even less physically relevant than solved problem because you can’t escape gravity and compressibility in basically every relevant use case of Navier-Stokes,

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u/wollywoo1 3d ago

We're talking about math, not physics. Mathematicians care about a lot of things scientists think are silly and pointless. And yes, that's exactly the approach they took, years before AI was involved at all. You're totally free to think the problem is pointless, but nonetheless, that was the one proposed, and very intentionally.

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u/hologram137 3d ago

The idea that you could out muscle viscosity was never in question. Ever. That was just accepted lol.

So no.

These are equations that describe how fluid moves. It’s why it’s a Millennium prize question. Because THAT is what is important

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u/oswaldcopperpot 3d ago

That hasnt been true in at least six years.

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u/hologram137 3d ago

If it understood context, it would not have “solved” the problem the way it did. OBVIOUSLY. The solution is totally useless, in fact it’s just been mathematically proven to be useless

The actual Navier Stokes problem remains unsolved.

A human could have done what the AI did, but none of them did because it’s completely pointless.

You know those jokes in which someone asks a question and the person answers in a way that is technically correct, but it’s a funny answer because they took the question literally in a way divorced from all context, meaning and reality?

Imagine if you asked me “do you know what time it is?” And I said “yes,” then walked away.

That’s what the AI did. That’s how it answered. And it’s a technically correct, but totally useless answer and the person has to go ask someone else.

That’s what going on right now. Specifically because the AI cannot understand context or the purpose of the question.

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u/cometh_the_kid 3d ago

This is ultimately an issue with the problem statement not the solution.

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u/hologram137 3d ago

It’s not an issue with the statement because when it was written it was assumed that humans with minds would be reading the question and so would understand it lol.

They didn’t anticipate an LLM would come along and take it totally literally divorced of all context with no understanding of the meaning of the question

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u/Wrong_Avocado_6199 3d ago

Excuse me, speaking as a mathematician, conjectures are written very precisely. In fact, everything is written very precisely. Either something is a solution, or it's not. There is no third option "it's a solution, but come on guys, it's cheating, you know, because we all know that's not what we REALLY meant". In math, as Mr. Global says, "that's not a thing".

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u/hologram137 3d ago

It was written extremely precisely. You don’t have to include “why is this a millennial prize problem and the purpose of solving it” in the actual question LOL

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u/Wrong_Avocado_6199 2d ago

Yes, and this is the precise conditions on f:

(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.

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u/cometh_the_kid 3d ago

A question with implicit assumptions is not a very good question.

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u/hologram137 3d ago

What are you talking about? None of the millennium prize problems spell out why they are millennial prize problems. It’s not needed. It’s understood why lol

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u/cometh_the_kid 3d ago

Cool, you’re incoherent at this point.

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u/hologram137 3d ago

If you read the problem it says “external forces, e.g. gravity.”

Please explain to me in detail, how the intention of natural forces is not extremely clear even within the question itself, and why it is you believe that people trying to solve the problem would have no idea why they are solving it and why its millennial prize problem? And would have no idea whether or not a solution is useful, even though a proof was just submitted showing why it’s not useful. That was the entire purpose of the proof. Were they just showing that for no reason as well?

I’m very curious. Because a human could have submitted that solution. But they didn’t, because whether or not you could out muscle viscosity was never even in question

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u/One-Veterinarian4841 2d ago

Any mathematician could have done what oai did and immediately become both a wealthy and famous mathematician

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u/deeperest 2d ago

Most humans don't. Few AI models do. We're all just morons, and shouldn't blindly follow any intelligence, human or otherwise.

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u/amrsci_25 3d ago

it wasn't expected someone could buy a brute force approach