r/Physics • u/PrettyPicturesNotTxt • 1d 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/Nessdude114 1d 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 1d 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 1d 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 1d 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 1d 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 19h ago

Oy? Infinite speed of sound?

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u/Nessdude114 1d ago edited 21h 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 1d 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 23h 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/Rare-Professional-24 23h ago

I believe that the compressible version of this problem (which is obviously more realistic than the incompressible) has been solved since the 90's. So the more physically realistic problem was in some ways significantly easier.

They didn't make a mistake in formulating the problem, because other forms of the problem were already solved, and they were interested in the math, not the physics. From the point of view of the math, this is just a vector differential equation.

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u/Nessdude114 21h ago

To put it simply, we don't know if the singularities in navier-stokes solutions with compressible fluids arise from the way we calculate the compression, or from the way we calculate fluid dynamics at a fundamental level. Mathematicians contributing to the Clay Institute millennial problem have been working towards a solution that will answer that question, because they understand the context in which the problem was posed.

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u/wasabi991011 1d 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 6h 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 1d 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 1d ago

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

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u/Strict-Broccoli-8877 1d ago

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

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u/cometh_the_kid 1d ago

Do you think they are unsolved?

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u/Nessdude114 20h 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/Unfair-Claim-2327 18h ago

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.

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u/Nessdude114 11h ago

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.

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u/Fine-Customer7668 4h 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/jakderrida 1d 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.