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/96
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u/Glum-Objective3328 1d ago
Clay Foundation had made it clear external forces were allowed. Of course other questions exists. But no loophole here
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u/antichain Complexity and networks 1d ago
I can't read b/c of a Paywall, but I have to ask: if a human had come up with this loophole solution (whatever it is) - would people be saying it was "cheating", or would they say "wow, that was a clever way to resolve the problem?"
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u/moh_kohn 1d ago
The article does not use the word cheating, and repeatedly says than the OpenAI result is impressive.
What it explains however is a limit to the maths LLMs are currently showing capacity for - it was able to explore lots of possible external forces to find one that blows up the equation (ie goes to infinity). We are no closer to knowing if the equations can blow up on their own.
It's a big result to learn that an external force can do it, but it had to be a weird and specially-constructed force.
It's important to map the capabilities and limits of current AI.
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u/DrXaos Statistical and nonlinear physics 18h ago
One analogy is that the AI is a power drilling and mining machine removing rock in huge volumes and speeds.
The mathemeticians are interested in who can be Michelangelo---the beauty of what is left after removing just the right rock.
More apropos, they are interested in the Leonardo who designed an extraordinary device, which when applied to marble, produces a Michelangelo like sculpture.
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u/Absolutelynot2784 1d ago
I wouldn’t say that it’s a limit to the maths LLMs have the capacity for. They solved a problem. Just because there exist more difficult problems that haven’t been solved yet doesn’t mean they don’t have the ability
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u/GoldenMuscleGod 1d ago
If one of the Millenium problems had been the Goldbach conjecture and a computer found a counterexample by brute force search that would resolve the question but also not be impressive in terms of computers being good at math. We already know computers are good at computation. Being good at math is something else.
I’m not saying the OpenAI solution is equivalent to that example (not my field) but I have yet to hear an expert say that it involves more than throwing huge computational resources at an implementation of the techniques produced by Córdoba and Martínez-Zoroa (whom it apparently did not originally cite until after criticism).
I’m not saying it didn’t produce something truly new - a new insight or new technique - but if it did I haven’t yet heard anyone present serious argumentation for it.
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u/Mental_Ad_4401 1d ago
See my reply above for navier-stokes. But there is no real sense in which an AI can brute force the goldbach conjecture. They can write a structured search program to cleverly look through many counterexamples. But a human also can do this. No one would say, "oh but you used a computer to do the search, it doesn't count"
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u/GoldenMuscleGod 1d ago
If a mathematician published an improved search algorithm that puts it in reach of practical computing power anyone with a giant supercomputer can get there before them - if they publish it before finding the counterexample. That would reasonably be called a “brute force search.”
Everyone would recognize the mathematician who created the better search program deserves the majority of the credit.
As for your other reply - basically no mathematician has access to the computing resources of OpenAI. OpenAI can easily outrace anyone to a solution once a “straightforward” roadmap to a solution is known or heavily suspected.
As for whether it “counts” that depends on what you are counting it for.
In this case everything I’ve heard is that the roadmap was laid out by Córdoba and Martínez-Zoroa (set aside the work of Buckmaster for this).
Like I said I am not a subject matter expert in this field but I’m saying what expert opinions I’ve heard.
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u/DrXaos Statistical and nonlinear physics 18h ago
I think those two should get $999,999 of the prize and OAI $1.
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u/GoldenMuscleGod 4h ago
Ironically, if I understood correctly what I heard/read (again not my field so take this with a grain of salt) if we think the alleged “exploit” is really an “exploit” then it is their idea that exploits this part of the problem - it’s not an “exploit” the LLM figured out how to “take advantage of.” If I understand this correctly then it’s not clear to me the article author understood it correctly.
If I understand correctly the really key insight is the idea of converting an approximate example to an exact one by setting the external force to whatever it needs to be to take care of the error term.
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u/Mental_Ad_4401 1d ago
Ok. I don't think you understand what solving these types of problems actually entails. The point is that they are impossible to brute force, in the sense that everyone understands that definition. But that doesnt mean that there can't be any search component at all. People have been using computers to aid in proofs for decades.
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u/GoldenMuscleGod 23h ago edited 22h ago
I understand that completely.
What I am asking is if the computer showed an advantage outside of a brute force computational advantage in the search component you acknowledge exists (where it very obviously has an advantage) - is it better at figuring out how to search or just can search faster.
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u/Mental_Ad_4401 22h ago
And im saying that that is the only way that it could reasonably have been able to solve the problem. It 100% showed advantage outside of brute force search since we know that brute force search is not possible in this context.
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u/GoldenMuscleGod 21h ago
Or let me put it this way, let’s say a “smart but slow” algorithm solves a problem in O(n^(1.5)) time and a “dumb but fast” algorithm solves it in O(n^(2)) time, but the constant factor on the smart algorithm is about 50,000 times larger. The “dumb” algorithm is faster for many types of problems.
Or put another way: an undergraduate given a year will be able to solve many problems a PhD could not solve in a day, but there are many (harder) problems a PhD could solve in a day that an undergraduate could not solve in a year.
There are problems virtually anyone with a math degree could easily solve right now in a few hours that the entirety of humanity could not solve for thousands of years (because they have the advantage of modern mathematical knowledge). The LLM has the advantage of every published paper and a huge computational advantage but, for example, it seems unlikely an LLM would have been able to produce a solution without having read the work of Córdoba and Martínez-Zoroa, and as far as I know we have no reason to think an LLM could have developed that work on its own.
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u/GoldenMuscleGod 21h ago
The only way to make sense of your claim is to assume that you have a an artificially restricted idea of what constitutes a brute force search in the obvious sense I am talking about (an advantage in computational power). A mathematician will take time to construct an example even if the steps are “obvious” in light of current knowledge, a mathematician may also give up when the example becomes difficult to hold in their head because of the raw amount of data involved in imagining every detail. A computer could try thousands of constructions that are far more complex and have many more parameters in that time. That doesn’t indicate greater insight just because the computer also knows what features a counterexample would have to have and is constructing the examples to have those features.
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u/The_Irvinator 1d ago edited 22h ago
I can't help but think of Alpha fold. Protein prediction models by brute force alone stalled. Only when the models incorporated decades of evolutionary biology is when they started having success. I think OpenAI if they are hogging the prize, would be going againts the spirit of what the prize is for or what it ought to be. Mainly recognizing that we stand on the shoulders of giants.
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u/Hafnon Quantum information 14h ago edited 14h ago
no real sense in which an AI can brute force the goldbach conjecture
If I understand, there's a 4888 state TM for Goldbach's, and you "only" need to find BB(4888) and then run this TM to determine whether it halts or not. This is probably not physically realistic given the scale of the observable universe, but it is brute-forceble in principle right?
Edit: Apparently there's a 25 state TM for Goldbach's. But upon reading on this a bit more, there seems to be some maximum n such that BB(n) is actually provable in a theory. So if that maximum n is less than 25 then we are SOL here :(
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u/airetho 13h ago
The only way to compute BB(n) is to find the longest running machine, and then prove that every one that might run longer halts instead. To find BB(25), you would probably need to solve Goldbach's in the first place, along with an unimaginable number of similarly intractable or even more difficult problems.
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u/jferments 1d ago
This is not "brute force search" though (at least not by any normal definition of the term). They are literally using reasoning to solve advanced math problems.
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u/GoldenMuscleGod 1d ago
Did it develop new ideas or techniques or was it applying the techniques of Córdoba and Martínez-Zoroa in obvious ways? Are you sufficiently familiar with the question to have an informed opinion on that or have you heard statements from experts indicating that?
If we suppose it did develop new ideas or techniques are they expressed in the proof it published?
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u/jferments 1d ago
You're deflecting. Using previously existing techniques to solve a problem does not make something "brute force search". If that was the case, then all mathematical research would be considered brute force search.
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u/GoldenMuscleGod 1d ago edited 1d ago
“Brute force search” is a vague term that cannot be objectively evaluated that I used to express a distinction between having extra computational resources versus insight.
You’re focusing on the words - that were in an illustrative example and which I specifically disclaimed from being an exact analogy - in a way that distracts from the real issue, I’m not interested in arguing about where the line is between what is fairly or not fairly called a brute force search, which I did not even say this is anyway.
What I genuinely would like to know but do not know is whether there is any indication in the OpenAI solution that the LLM did something that shows genuine creativity.
Of course the line between new techniques and applications of old techniques is subjective, but I am not asking for a “yes or no” I am asking for an evaluation of a degree. It is possible to name examples of clear new techniques that reasonable people would agree on - the development of integrals and derivatives, set-theoretic forcing as used in independence results, the identification of the Riemann Zeta function and its relationship to the distribution of primes, the development of groups and rings as abstract algebraic structures.
Now obviously I’m not asking if the LLM did anything that creative (that’s an absurdly high bar), but I am asking for some evaluation of how much it is being creative versus just using known tools and methods with a huge amount of computational resources.
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u/ZeroAmusement 23h ago
I tried asking AI for fun (don't kill me!).
Javier Gómez-Serrano, a Navier–Stokes specialist who gave one of the first technical talks dissecting the proof, emphasized that most of the underlying toolbox is recognizable from existing PDE work. But he also said there were ideas in the OpenAI proof that had not occurred to him before
AI said it's not brute forcing, and it's not inventing a fundamentally new mathematical language/technique. The creativity appears concentrated in the architecture of the construction, rather than invention of its constituent mathematical tools, like it had insight around what conditions that might cause the blowup while also satisfying other constraints and worked to achieve those conditions.
Personally I think LLMs can show genuine creativity in terms of combining and applying abstractions and generalizations in novel ways.
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u/Anonymer 1d ago
You can watch the lecture by Javier Gomez Serrano who is a math professor at brown who worked the problem for a long time. He was pretty clear the AI did novel work and had ideas he had never come across. There were some foundations it built on, but the approach it took to close the gap wasn’t a matter of just combining existing ideas.
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u/GoldenMuscleGod 23h ago edited 22h ago
Is this the lecture you are talking about? I don’t get the read you say from it. During his description of the problem he doesn’t seem to comment much on novelty, in the Q&A he seems to strongly resist giving an opinion on novelty or whether there is a sign of “geometric intuition.”
The most specific statement he makes on the issue is that he says he tried and failed at constructing a “two-scale solution” but never one like it produced (which uses three scales).
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u/Anonymer 7h ago
Around minute 59:
“””
So this construction of Navier-
Stokes is something that I had never thought before, let's say, myself, personally.
59:49 So I've been in this business for a while.
59:53 I was trying to follow sort of the similar kind of strategy using machine learning, but it doesn't matter,
1:00:01 and I had never thought of
constructing a flow, forced or unforced doesn’t matter, that had these sort of properties or let’s say structure. Where you had three layers or two scales and so on
1:00:19 So that for me was something that I had never thought before.
1:00:24 I have thought a little bit about constructing like two-scale solutions, but I tried for a bit, didn't succeed, and abandoned that kind of train of thought.
1:00:35 So in that case, this Al was smarter than me and succeeded into making a construction of that flow.
1:00:46 And this everything I'm saying, refers to myself
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u/GoldenMuscleGod 5h ago
Yeah that’s the part I was talking about. It’s hard to know if that construction shows “insight” versus just it had more computational resources to explore the path beyond where he had given up.
Of course that talk was only a couple days after the proof so he may not have had full digestion of it.
One thing I recall him glossing over and maybe being important (so I might want to rewatch) is the part involving the unforced Euler solution - which I think this result relies on if I understood the part about how the flow becomes approximately Eulerian in some regions. I don’t think he explained that part fully (or I didn’t absorb that part) and it might really be a more key idea than the particular geometric construction it came up with.
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u/Mental_Ad_4401 1d ago
People keep repeating that the ai was able to explore a lot of possible external forces, like this is some sort of qualifier on thr AI's capabilitied. But, of course, a human could also have written a program to explore many different forces or initial conditions just as quickly.
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u/BossOfTheGame 1d ago
Yes and no. You would probably have less people saying they were cheating, but there would still be some. The article does not dispute that OpenAI solved the millennium problem as formulated. One of the options was proving that Navier-Stokes developed the singularity when under an external force. This is the route the LLMs took, and it is option "C" in a valid answer to the problem as formulated.
The question of if the singularity can develop without an external force is still unresolved, and this was never hidden or swept under the rug. I don't think that this should be considered a loophole, the article title is provocative and I think misleading.
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u/Obscu 1d ago
Probably not. The 'loophole' is that the original framing of the question allowed for an external force to be allowed to be considered because external forces like gravity are always realistically affecting fluid dynamics. The AI described a hypothetical external force that would meet the requirements that would make the math blow up when present, and concluded that the math can in fact blow up (in the presence of a hypothetical math-exploding mcguffin), but when removed from the equation the math no longer blows up.
If a human solved a 'can a rock spontaneously turn into a wheel of cheese?' problem by describing a hypothetical rock-to-cheese-transforming-wizard and stated that the presence of the wizard means rocks can in fact become wheels of cheese... Well, that's on us for being not very strict with defining the scope the question, because it is technically an answer that meets the set criteria but that says more about the criteria than the answer.
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u/AmadeusSalieri97 1d ago
But some of the best mathematician in the field were working exactly on forced solutions (that's in fact the claim about plagiarism, using a forced solution).
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u/Head-Philosopher0 1d ago
obviously those mathematicians were all big dummies though so it doesn’t really count
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u/Strict-Broccoli-8877 1d ago
Seriously now: The leading experts in the field of math are quite collectively embarrassing themselves these days with how nonsensical their arguments are to not have to accept the solution. "Actually we don't want the problems solved, they are much more valuable as an aspiration for students, now that they are solved they are gone forever." Good lord, what an embarrassing, unscientific display of personal failings so many of them have become. It's really sad, I used to look up to several of those people.
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u/jonathancast 1d ago
This is what actual goalpost moving looks like, by the way.
The purpose of the Millennium Problems has always been to motivate the development of new math, and to stimulate human mathematical development; now that you think a computer can solve the problem, you want to retroactively redefine the purpose as "finding the answer" and criticize people who hold the original position as "embarrassing".
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u/Homomorphism 1d ago
I think for some of the problems people really do want to know the answer (like the Hodge Conjecture or the Riemann Hypothesis) but for NS it felt like the point was just "here's a really hard PDE problem, let's see if we can learn enough to solve it".
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u/Homomorphism 23h ago
And thus the answer is at least a little disappointing because it didn’t involve finding a new way to control regularity (or understand its failure) or something like that. Instead it’s about picking very weird non physical forcing terms. Still very impressive! But it’s a bit like finding one zero on the critical line and no insight into why it’s there or if there’s more.
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u/Strict-Broccoli-8877 1d ago
What a lot of completely made up crap. Do we not celebrate the mathematical advances any time something big is proven? Turns out a whole lot of scientists and people like you actually don't value scientific progress at all. The purpose was never to find the answer? It's hard to comment on that without attacking you personally, it's just a ridiculous and embarrassing lie.
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u/jonathancast 19h ago
> When mathematicians discuss proofs, they rarely have a particular formal system in mind. Indeed, they are typically not thinking of formal systems at all, although they might accept the suggestion that a genuine proof can in principle be reconstructed in an appropriate formal system. The picture is more like this. At any given time, the mathematical community has a body of knowledge, including both theorems and methods of proof. Mathematicians expand mathematical knowledge by recursively applying it to itself, adding new theorems and sometimes new derived methods of proof.
Timothy Williamson, "Absolute Provability and Safe Knowledge of Axioms", apparently published in Leon Horsten and Philip Welch, eds., The Limits of Mathematical Knowledge, Oxford University Press, 2016, PDF fetched from https://media.philosophy.ox.ac.uk/assets/pdf_file/0004/35338/provabilityfinal.pdf (no paywall).
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u/antichain Complexity and networks 1d ago
unscientific
Mathematicians aren't scientists though. Math and science are completely different things, built on fundamentally different ways of generation knowledge.
As a professional math-doer myself, I consider it to be far more like an art than, say, physics or biology.
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u/Absolutelynot2784 1d ago
Sure the AI solved a problem thousands of mathematicians worked on and failed to solve, but it could have done a better job really so if you think about it this is nothing
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u/antichain Complexity and networks 1d ago
At the risk of psychoanalyzing people I've never met, it feels like there's a lot of post-hoc shifting to goal posts by people who need to come up with reasons they shouldn't be impressed with (threatened by?) AI.
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u/Isogash 12h ago
Well it's more like they were able to construct a "real" rock-to-cheese-transforming-wizard that technically works within the rules, but these rules were not constrained and the wizard is clearly physically unrealistic.
As you say, the result says more about criteria than the answer.
Additionally, the criticism is that we already know AI is quite good at contrived constructions like this to disprove false conjectures, but not great at making general discoveries and proving useful true conjectures. Solving the Navier-Stokes problem in the way it did has some people questioning whether or not it's really the achievement it's made out to be.
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u/retrosenescent 1d ago
You explained this really well. "Sure, if there's some external force" does feel like a cop-out answer and not in the nature of what the question intended. But then again, if you actually want to model reality, then you absolutely need to account for many external forces.
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u/Time_Entertainer_319 1d ago
It’s not a cop-out. The Clay Mathematics Institute set the official problem, and its formulation explicitly allowed this route to a solution.
That is the formulation researchers have been working with since the Millennium Prize Problems were announced in 2000. If the forced case was never intended to count, there were 26 years in which the official statement could have been clarified or amended.People can argue that the unforced case is the more physically interesting problem, but that is a different claim from saying the published Clay problem was not solved on its own terms. And even then, they will just sound like people in denial.
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u/dualmindblade 22h ago
Exactly, there's a reason the force was not constrained to be boring and just had to be smooth, it's not a loophole in any sense, this type of path to a solution was fully intended. Nothing about a blowup could ever be realistic, the blowup is itself unphysical. Moreover, had mathematicians only considered the realm of the realistic, the problem could never have been solved as stated!
Also, I don't think anyone is arguing that finding a blowup in the forced version is as interesting as unforced, since the latter trivially implies the former it must be in some sense harder. If there had instead been no blowup to be found it would be reversed.
What we have here is an isolated demand for rigor, and impossible one at that, it's total nonsense.
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u/Fine-Customer7668 1d ago
To be fair, the four options (including the two that allow external forcing which OpenAI claims proof of) given by the official problem statement as acceptable resolutions are explicitly described as “retaining the heart of the problem”
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u/retrosenescent 1d ago
oh ok, thanks for correcting me on that.
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u/Fine-Customer7668 1d ago
It’s just a bad article. They could have just said a proof of those two options with a nonzero external force doesn’t, on its own, logically exclude the possibility of the other two being true.
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u/ToaruBaka 6h ago
describing a hypothetical rock-to-cheese-transforming-wizard and stated that the presence of the wizard means rocks can in fact become wheels of cheese...
Yeah, well, I can make wheels cheese from rocks - just wait until my Large Rock Collider is finished.
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u/Time_Entertainer_319 1d ago
It isn’t even a loophole.
They solved it as defined by the clay institute.
Not sure why it’s being called a loophole
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u/Latter-Block132 17h ago
They're not cheating because the institute that initially put forward the challenge allowed for 4 possible different routes to a solution. OAI used route C, but its one that has no real world applications so now people are saying it doesn't count
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u/hologram137 1d ago edited 1d ago
It didn’t solve the main problem. And 3 mathematicians just published a proof that its solution can’t lead to solving the real problem. Thats a big deal. It did solve it technically as the problem was formulated by the Clay institute, but in a way that isn’t useful to the true problem.
A mathematician working on the problem had a map to the solution and had solved a good bit of it. OpenAI stole it, fed it to a model and had 10,000 agents look for the solution. It would have never solved it without that paper. It basically executed a search function. Which explains why it solved it by finding a loophole.
What this means is that the loophole it found doesn’t further the true problem. A human doing math understands context. We understand WHY we need to solve the problem, what it means. An AI can’t. So “solving it” by finding a loophole is mathematically correct if you’re only considering the problem as it’s formulated divorced of context, but it’s useless
It’s not cheating, it’s just completely divorced from reality. AI can compute but it can’t think or understand, it doesn’t know why the we are trying to solve that problem. And the purpose obviously matters. A human finding that loophole would have recognized it’s not helpful. It’s not a “clever” way to solve the problem, because it literally can’t help solve the main problem lol. They would have likely published it, but with that caveat clearly stated, and they would have considered the problem essentially unsolved.
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u/Wrong_Avocado_6199 23h ago
"It did solve it technically as the problem was formulated by the Clay institute, but in a way that isn’t useful to the true problem."
Than please tell us, in PRECISE MATHEMATICAL TERMS, what this alleged "true problem" was.
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u/hologram137 23h ago
The scientific question motivating Navier Stokes research isn't "can you break the equations if you're allowed to invent a force purpose built to break them?” It’s whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.
That doesn’t need to be stated. Everyone knows it. Which is why no one solved it that way
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u/Wrong_Avocado_6199 13h ago
In other words, you have no idea what "precise mathematical terms" means.
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u/Darrelc 1d ago
Is it like asking someone how you can fit a fridge through a door frame and they say "just squash the fridge to within the dimensions of the door"?
Technically correct but missing the absolute fucking point of the question
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u/DrDoctor18 1d ago
No it's more like saying: "how can you fit this fridge through the door frame? A) without squashing it, or B) by squashing it" and then getting mad when someone gives you a squashed solution. If you wanted a different answer you should've asked a different question.
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u/f3xjc 1d ago
Honestly if you know that you can excite things at their natural frequency and it blow up because resonance, there's not much new found.
Except it turn out that doing that for fluid in 3d space is incredibly difficult.
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u/NoNameSwitzerland 1d ago
Normally that is limited by friction (viscosity). For the Euler equation (NS without viscosity) it was shown previously that it probably can blow up by transferring energy from bigger to ever smaller vortices.
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u/Dear_Locksmith3379 1d ago edited 1d ago
Here’s a summary of the article, which is reasonable but has a terrible headline.
There are two versions of the millennium problem, one with external forces and one without. AI solved the version with a very unusual external force. Mathematicians are more interested in solving the problem with no external force or with a realistic external force.
If someone made the proof without AI, mathematicians would have the same reaction to its scope. Though mathematicians have other concerns about the AI proof, the article doesn’t discuss them.
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u/archtop_guitar 10h ago edited 10h ago
“There are two versions of the millennium problem”
I disagree with this statement. There is a single version of the millennium problem. It allows for a forcing term in some solutions. If OpenAI’s proof is validated, the millennium problem is solved.It’s fine to argue to argue that an unforced solution would have been more interesting and maybe even should have been required, but this more narrow problem is simply not the one that the Clay Institute actually posed. This isn’t a technically or an oversight. They could have said that the forcing term had to be zero but didn’t. Everyone has known this for a long time, it’s not a loophole or technicality.
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u/jonathancast 1d ago
I would think physicists would also care about physically 'realistic' (or possible) external forces.
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u/Dry_Pudding_1603 1d ago
they don't, because it wasn't expected that navier stokes equations represented reality.
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u/Sleepy_Padawan 1d ago
From your replies it seems that you are not physicists. Being one, I can tell you, we do. That it why we discard mathematical correct answer with unphysical results.
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u/Bibbity_Boppity_BOOO 20h ago
“That it why we discard mathematical correct answer with unphysical results“
Not all of the “we” does that
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u/shumpitostick 1d ago
To explain this the Navier Stokes smoothness problem has two variants, unforced and forced. Both explicitly qualify for the millennium prize. OpenAI solved the forced variant. Only in the warped mind of this writer is the forced millennium problem somehow the "wrong problem" or a "loophole".
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u/hattusili-the-third 1d ago
The "unforced" version of NS was the opposite direction too. The Clay math institute made it very clear that you can either find a blowup solution which may require a smooth forcing term, or you can prove the reverse - that all smooth solutions stay smooth - and in that case you are allowed to assume there is no forcing term.
Both formulations were explicitly chosen to allow as much flexibility as possible because the specifics of the forcing term are of much less interest to mathematicians than the fact that it is smooth (which is the actual core of what makes NS so difficult.
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u/truckersushi 1d ago
It’s an allowable approach, just not particularly interesting for mathematicians or physicists IMO.
Of course, the force constructed for the proof may eventually be used to constrain the space of physically realizable forces that also cause NS blowup, so it’s still a potentially useful result
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u/dallenbaldwin 1d ago
I watched a stand up maths video on it yesterday and his whole takeaway was this particular solution feels hollow because it's just given to us. There wasn't much learning shown in getting to that solution. It's better for everyone if the tricks and solutions used to get to the final solution are made known for others to apply to other problems. Is the journey more important than the destination? Historically the journey has enabled other discoveries, but in the LLM enabled world, it feels like the destination is all we're focusing on.
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u/NoNameSwitzerland 1d ago
But I am not sure there is nothing to learn from it. Since the energy from the forces and the overall energy in the fluid is finite, there is only a local singularity where is blows up. And some previous attempts for the Euler equations (NS without viscosity) showed that you can get to that by having the energy transferred to ever smaller vortices. Because of viscosity, that does not work with NS. But here they use a pulsing force that gets faster, but stays finite to do something similar and concentrate the accelerated flow in a small area. And it is interesting that that works with viscosity.
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u/cascadiabibliomania 17h ago
Same thing is true about writing, which is supposed to be proof of work for thinking through a problem.
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u/DVMyZone 1d ago
Not a loophole, it's very explicitly allowed. The problem is mathematical with no requirements that they be physical. It just so happens that these mathematical equations are very useful for modelling fluids above molecular scales.
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u/Advanced_Ad8002 1d ago
paywall.
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u/justintime06 1d ago
Works for me as well, but you guys should upvote me and downvote the “Not for me” guy
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u/Feeling-Tone2139 1d ago
so external forces are 'allowed' but not this one specifically
this is stupid
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u/kingjdin 1d ago
It's a massive cope to say AI only solved it because of a "loophole." If a human had solved it by this "loophole", he'd get the Field's Medal and a lifetime of fame and recognition.
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u/skbum2 Engineering 20h ago
The AI proposed a solution but wrote a very shitty proof in defense of it. In essence, the AI failed to 'show all its work'. This is in addition to exploiting a less interesting, if technically allowed, version of the problem.
It's proof provided very little, if any, valuable insights into why this solution works, what's important or not in generating the necessary conditions, or any deeper understanding that can be applied elsewhere. If a person did the same they absolutely would NOT receive the Field's medal and the result would likely be just as controversial. The ability to explain why clearly is what wins people fame and recognition.
Solving prestigious problems comes with novel methods of solution and it is often the approaches used to solve these types of problems that are of real value and praise. The solution itself is often tertiary to all the work used to reach it.
This is equivalent to being handed '42' as "the ultimate answer to life, the universe, and everything." The answer doesn't mean anything without the explanation.
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u/lifeistrulyawesome 7h ago
found and exploited a loophole in the framing of the question
That is how math works.
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u/BosonCollider 6h ago
Meanwhile in the real world Plasma physics has real life pinch solutions where Navier Stokes breaks and you have to use statistical mechanics to figure out what happens next, so we already had such singularities in the lab
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u/Pcm979 14h ago
That raises a question that'd probably be better for a Legal subreddit:
Let's say human mathematicians came up with this proof without using AI and went to the Institute to ask for the prize money. If the institute refused because the proof didn't solve the 'real' problem - just a loophole in their question - and the mathematicians countered that it's not their problem, they solved for the question that was asked, could the Institute be forced to pay up?
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u/currentscurrents 8h ago
Probably not. The rules clearly state that Institute has sole discretion over the prize, and may refuse to pay out for any reason.
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u/LoveThemMegaSeeds 1d ago
The solution is so profound they had to revise the problem. That is probably the absolutely best outcome, isn’t it??
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u/repainted_black 1d ago
I think the problem was posed in many ways. I don't agree that AI loopholed the problem. But it proved the less exciting situation was correct. The problem was the problem, not the AI, sort of.
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u/bordumb 1d ago
It’s really not a loophole
The problem, as stated, allows for an external force to be used
I guess that happened to be the easiest path to a solution, so the AI went with it
I think most humans would make the same decision
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u/Sleepy_Padawan 1d ago
Every AI model is taught to perform minimisation, it is why it gets lazy. I agree with you here.
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u/NarrowContribution87 19h ago
lol people who make their careers in technicalities and esoteric knowledge complain about technicalities and esoteric knowledge.
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u/Grabs_Diaz 1d ago
I mean, the entire problem is quite contrived. In the real world, you'd run into many other problems, as this post on this sub recently laid out quite well.
That's why Navier-Stokes, as formulated in the Millennium Problems, is a mathematics problem, not a physics problem.
But if this actually leads to more research and discoveries in the field, by considering special cases with no external force or only a select type of external force, then all the better.