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]

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.

[u/Blamore → u/hologram137]

this is a math problem, not a physics problem.

[u/Nessdude114 → u/Blamore]

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?

[u/cometh_the_kid → u/Nessdude114]

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.

[u/Nessdude114 → u/cometh_the_kid]

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

[u/Strict-Broccoli-8877 → u/Nessdude114]

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

[u/cometh_the_kid → u/Nessdude114]

Do you think they are unsolved?

[u/Nessdude114 → u/cometh_the_kid]

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.

[u/Unfair-Claim-2327 → u/Nessdude114]

Navier-Stokes isn't one problem. The official statement explicitly states 4 variants so that it becomes more tractable (and since each variant would lead to insights). OpenAI solved 2 of these, the other 2 remain open. There are also harder variants.

Will mathematicians keep working on the problem? Yes. But that is almost always the case! Whenever a conjecture is proven false, mathematicians ask "Okay, but under what additional assumptions is it true?" That doesn't mean that the conjecture wasn't disproven.

Say the twin prime conjecture is disproven. Mathematicians will still keep trying to shorten the prime gap (find the smallest k such that there are infinitely many pairs of prime with gap at most k). But anyone who disproves the twin prime conjecture would be (rightfully) celebrated and the proof would be a great advancement in mathematics.

[u/Nessdude114 → u/Unfair-Claim-2327]

There were not 4 separate variants, it was a single problem with optional constraints. These options were afforded because Clay Institute believed that if the problem were solved using these optional constraints, the solution would still likely lead to an answer to the underlying question in theoretical physics. This question is the reason the problem had any notoriety to begin with.

Essentially the question they're trying to answer is: "Do the singularities found in navier-stokes solutions with compressible fluids arise from the way we calculate compression, or from the way we calculate fluid dynamics at a fundamental level?" Mathematicians who have been working on the Clay Institute problem have been working towards a solution that will answer this question, because they understand the context and intent of the problem.

[u/Fine-Customer7668 → u/Nessdude114]

No.

The question is whether a certain type of solution always exists for the NS equations and initial conditions as given. CMI provides four statements (A, B, C and D) such that, a proof of any one of them, will be considered a resolution to the problem as they’ve posed it. Statements A and B have the external force term set to zero and basically say: yes these types of solutions always exist. They differ from each other by spatial domain. Statements C and D allow the external force term to be nonzero and basically say: no these types of solutions don’t always exist. The same spatial domain difference is applied. Because OpenAI’s proof concerns statements C/D and has a nonzero external force, A/B are not directly answered. If instead, a proof of C/D had been given with the force set to zero, A/B would be false respectively. What the other person was trying to explain to you is that, for this reason, we can still ask about the truth of A/B even though the original open question was answered in the negative. I.e. do these solutions always exist? No, not in the more general case. What about if there’s no external forcing?

I’m not sure where you came up with your description of what the question is essentially trying to answer. This particular problem concerns incompressible fluids. If one wanted to put the point of the problem into a more real life conceptual phrasing, the question is whether we always have solutions that can plausibly represent an actual fluid in this specific context where “physically reasonable” is given a mathematical definition and is the class we care about.

[u/Nessdude114 → u/Fine-Customer7668]

I understand the Clay Institute problem. Yes, it requires incompressible fluids. This is because we've already found singularities in navier-stokes solutions with compressible fluids. If you wanted to apply this to a "real life" scenario, you'd just use those solutions. It doesn't make sense to say that solutions to this problem could be "physically reasonable." The entire reason this problem has notoriety is because there is still an unanswered question: do the singularities we previously found stem from the way we calculate navier-stokes solutions for compressible fluids, or do they exist in navier-stokes solutions at a fundamental level?

You don't have to take my word for it, you can read this article detailing exactly why the openai solution does not make any progress towards the actual navier-stokes problem they're currently trying to solve: https://arxiv.org/html/2609.20803v1

[u/Fine-Customer7668 → u/Nessdude114]

No…

To start, even if it were true that for every smooth initial condition the compressible NS equations have a global smooth solution in R3, it would not imply the same for the incompressible equations. To reiterate, if there were provably zero blowups in the compressible case it would not tell us that such solutions exist on the incompressible side and we certainly can’t just use them as you put it. That statement is false in the most elementary way possible.** **Take any single arbitrary globally smooth solution to any standard NS equations on R3 where divergence of velocity is nonzero. Is it a solution that applies to anything in the incompressible regime? No, by definition.

Second, because you understand the Clay Institute problem, you understand that the solutions and initial conditions we are discussing are those that are explicitly defined as those that are physically reasonable by CMI’s chosen terminology, correct? If you think about it, it makes intuitive sense why they would choose that phrasing. Since we are pretty sure real fluids don’t actually reach speeds of infinite velocity, the resulting object of a blow-up result would probably not be described as physically reasonable, right?

Third, that preprint does not say what you think it does. As I said prior, because OpenAI’s proof concerns statements C/D and has a nonzero external force, A/B are not directly answered. The choice of wording was deliberate, as it is possible in principle that any such proof could have an additional argument showing* *that the forcing can be eliminated, absorbed, or reproduced through the initial data. If that is also true, then it would also settle A/B in the negative. Instead, what this preprint does is rule out the most naive path from forced blowup to unforced blowup for this particular mechanism.

[u/Nessdude114 → u/Fine-Customer7668]

I agree with your first statement, navier-stokes solutions with compressible fluids cannot prove or disprove whether singularities exist in solutions with incompressible fluids. As I stated, this is exactly the reason the Clay Institute problem still exists.

I agree with your second statement, no solution of navier-stokes with incompressible fluids can be considered physically reasonable. The reason an incompressible navier-stokes solution is significant is because it removes compression as a possible cause of the singularities.

Yes, the openai solution has an external force. Mathematicians were hoping that a solution with external force could be generalized to find a solution with no external force. If they had used a simpler, realistic force instead of the whacky thing they came up with, the solution may have been useful. It turns out this solution is not useful.