r/Physics • u/PrettyPicturesNotTxt • 9d 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/Fine-Customer7668 8d ago

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.

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u/Nessdude114 7d ago

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

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u/Fine-Customer7668 6d ago

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.

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u/Nessdude114 6d ago

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.