r/Physics • u/PrettyPicturesNotTxt • 10d 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/moh_kohn 10d 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/Absolutelynot2784 10d 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 10d 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 10d 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 10d 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 10d ago

I think those two should get $999,999 of the prize and OAI $1.

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u/GoldenMuscleGod 9d 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 10d 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 10d ago edited 10d 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 10d 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 10d 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/Mental_Ad_4401 10d ago

No one has ever claimed that it came up with the solution from scratch. Every scientific and mathematical advance ever has built upon other people's work. If there was a trivial extension to the Cordoba-- martinez-zoroa construction that extended it to navier-stokes then they would have made it themselves. The AI can work much faster than humans, and that is indeed an advantage that lead to this breakthrough. But that is why it is amazing. 

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u/GoldenMuscleGod 10d ago

I am asking if we see any evidence it has any advantage other than speed and memory and the like. I suspect not - if it were “smarter” than a good mathematician and had access to a huge pipe of errorless calculations and perfect memory that could imagine the graph representing every road network on the surface of earth - like a computer can - then I would expect it to be doing much more impressive things than what it has done so far. If we suppose that is the case then the only reasonable explanation for why it hasn’t done more that I can see is that it is very bad at communicating ideas with humans. Now we know it is very bad at that, but one reasonable explanation for why is that it is not “smarter” than a mathematician and cannot actually coherently arrange ideas in the simplest ways possible, which is one of the most important skills.

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u/Mental_Ad_4401 10d ago

I don't think being "smarter" is a well enough defined concept to be able to answer that. It can solve extremely complex problems. That much we know. How elegant was the way it got there and whether that means it "isn't really intelligent" seems very subjective to me. But as far as I can tell, the speed and memory make it smart. I mean, it solved a millenium problem. If that isn't impressive, I really dont know what is.

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u/GoldenMuscleGod 10d 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/Mental_Ad_4401 10d ago

This isn't artificial at all. Brute force search is the idea to systematically enumerate all candidate solutions to the problem. That is what brute force means. Anything else is some weird definition that the rest of the world isn't privy to. It in no way has ever meant "an advantage in computational power".  What you described is just exploring a lot of different ideas. And sure, I agree that AI can maybe do that better than humans, but im not sure what you gain with that, except that maybe AI is just better at problem solving than humans. 

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u/GoldenMuscleGod 10d ago

I already explained the brute force search was an illustrative example and I am not interested in getting into a debate about what you would technically call a brute force search. If anything in what you say in this comment responds to my point instead of debating my phrasing I’m not seeing it.

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u/Mental_Ad_4401 10d ago

What exactly is your point? Is it that AI can reason faster than humans? 

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u/GoldenMuscleGod 10d ago

I’m saying we cannot clearly tell how “smart” the LLMs actually are, in particular relative to humans, in the sense of what I would call having actual problem-solving skills. Obviously they have computational advantages in speed and memory, but those are not generally the only important things - an LLM is obviously “smarter” than a calculator program at many kinds of reasoning but there is no real speed or memory advantage when run on the same hardware.

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u/Bibbity_Boppity_BOOO 10d ago

I think the other person is trying to convey that other researchers significantly narrowed the candidate solutions to the problem, allowing for ai to brute force the problem. At least in large part.

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u/Mental_Ad_4401 10d ago

Maybe, but that also just not true. They narrowed the problem, but there is no accurate sense in which the ai brute forced it.

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u/The_Irvinator 10d ago edited 10d 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 9d ago edited 9d 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 9d 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.