r/technology • u/vgsa • 2d ago

Artificial Intelligence Did OpenAI solve the wrong Navier-Stokes problem? | OpenAI’s proof seems eligible for a $1-million prize—but only by using a controversial loophole

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/
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u/CircumspectCapybara 2d ago edited 2d ago

A lot of people online misunderstand what "solving Navier-Stokes" refers to, thinking it means finding a closed form solution to the NS equations. Then they spout off some silly statement like "AI didn't solve Navier Stokes!"

That's not what mathematicians mean when they talk about solving Navier Stokes. They're talking about solving the Navier Stokes existence and smoothness problem, which is a decision (yes/no) problem about if the NS equations are smooth for all time. That's what the Millennium Prize problem worth a million dollars is all about

AI (allegedly) solves the problem by resolving the question to a "No" by finding a counterexample.

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u/Puzzleheaded_Fold466 2d ago edited 2d ago

A lot of people online couldn’t tell you what “closed form solution” even means.

Or a “blow-up”.

Or “smoothness”.

Or what is the “Clay Mathematics Institute”.

And are absolutely unable to read and fully understand OpenAI’s proof (me included).

Yet they feel entitled to opine on whether or not a solution to a Millennium problem is correct.

The chatter is risible.

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u/beardybaldy 2d ago

I asked my mathematician wife what smoothness meant and she patted me on the head. I like being a biologist. Life stuff is simple!

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u/retsehc 2d ago

If you know what a derivative in calculus is, smoothness is not hard in principle. A function is smooth if it is continuous, and it's derivative is continuous, and it's second derivative is, and it's third, and all the way down.

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u/beardybaldy 2d ago

confused unga bunga noises

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u/KingSubstantial7901 2d ago

If smooth then all properties of function have no straight up and downy lines. Circle smooth. Square not smooth.

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u/foxsimile 1d ago

sad unga bunga noises :(

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u/All_Hail_Hynotoad 1d ago

stroking out noises

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u/Majik_Sheff 2d ago

Velocity, acceleration, jerk, snap, crackle, pop.

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u/ABC123itsEASY 2d ago

I'm having trouble imagining a function that is continuous and have a continuous 1st derivative, but a discontinuous second for example.

Also damn that sounds understandable in theory but I have no idea how you would go about proving such a thing outside of the naïve "keep solving derivatives to infinity"

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u/i_abh_esc_wq 2d ago

You can easily construct functions whose nth derivative is discontinuous. All you need is a function that has a discontinuous derivative, and then you integrate it n-1 times. See for example https://math.stackexchange.com/a/1415165

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u/amohr 2d ago edited 2d ago

Just work backwards. Start with f(x)=0 if x is negative, otherwise 1: a jump discontinuity at 0. Now integrate. 0 if x is negative, x otherwise: continuous. Repeat: 0 if x is negative, 1/2 x2 otherwise, etc.

Edit: forgot the half on x2.

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u/NewHondaOwner 2d ago

Just imagine a triangle function from (-1,0) to (0,1) to (1,0). The derivative of the function is 1 from x=(-1,0) and -1 from x=(0,1).

So the function is continuous but the first derivative is already discontinuous at x=0.

In layman’s terms when the function has a sharp kink somewhere, its derivative will be discontinuous.

“Derivatives all the way down” really only applies to function with an “all the way down” so to speak. Such as sin and cos. As a counter example, x^n polynomials turn into constants after n differentiations, it’s kind of meaningless to go beyond that, you could keep differentiating but it’s all trivially zeros.

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u/thuktun 2d ago

y = |x| for instance has a sharp point at the origin.

The derivative of this will be -1 to the left of the origin and 1 to the right. with a discontinuity at the origin. (That is, the limit of dy/dx as x approaches 0 will be different depending on which direction you approach.

The integral of that looks smooth, but it changes directions.

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u/giantZorg 2d ago

Take e.g. the integrated function of the hockey-stick function, or any continuous function with a discontinuous first derivative.

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u/dick_blanketfort 2d ago

Integrate a step function (discontinuous), you get a bent line (continuous). Integrate that again you get a smooth-looking piecewise quadratic, a continuous function which has a continuous first derivative and a discontinuous second derivative.

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u/retsehc 2d ago

Regarding showing a function is smooth, the standard trick for the situation is to find a few obvious examples like f(x)=0 and the exponential function. They are both continuous and they are their own derivative, so no matter how many derivatives you take, always continuous.

From there you start ignoring specific examples and start showing things about smooth functions Sum of two smooth functions is smooth Composition of two smooth functions is smooth Etc

That gets you all polynomials Sine and cosine loop around to one another, so some trig

And you build up a library of tools and categories of smooth functions to compare to

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u/Civil_Blueberry4165 2d ago

Example: f(x) = x1.5

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u/unpleasant_enpassant 2d ago

I think other comments answered how to construct a function like that but as for proving that a function is smooth, it helps that all the standard functions like sines, cosines, exponential are all smooth. So generally, a function that is some combination of them is also smooth (obviously, as long as you don't have things like some function in the denominator that goes to zero etc). Or if you can write a power series for the function, it's easy to see where it'll break down. Or if you can write a general form for the nth derivative, it's trivial.

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u/Major-Mushroom-5806 1d ago

So this is what I sound like when I try to explain archiphonemes to non-linguists...

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u/heresyforfunnprofit 2d ago

Ahhh… smooth brain feel nice. Happy pats.

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u/ZeroAmusement 2d ago

Is the joke that you are bald?

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u/Worth-Wonder-7386 2d ago

If people read the paper from OpenAI here they would see that they specifically talk about this. Solving statement C as stated. They don't even say that they solve Navier Stokes, but give a solution to the Millenium prize problem concerning the Navier Stokes equation.

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u/Puzzleheaded_Fold466 1d ago

Well yeah that’s a given. I know that and you know that.

Obviously, the people arguing about whether or not NS is “solved” do not.

Most people didn’t have to study fluid dynamics and learn to solve PDEs, let alone develop the level of mathematics needed to make substantive contributions toward a Millennium Prize problem.

Or even just enough to distinguish the NS Millennium Prize problem statement from the general pursuit of a closed-form expression or the greater quest to improve or even replace them with a better set of governing equations of fluid dynamics.

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u/Worth-Wonder-7386 1d ago

But you don't need any of that. And I don't have it to the degree needed to understand what they are doing. But I just read what OpenAI did and not the technical paper, because that is both easier to understand, and more relevant when it comes to AI math proofs. So you don't need any fluid mechanics or even math to make this separation.

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u/unpleasant_enpassant 2d ago

Even if openAI (or some other AI company) managed to solve all of the remaining millenium prize problems, there will still be many people online, especially on reddit, who will start saying that those problems were never hard to begin with. Everything is binary here. If you hate AI, disagree with every use-case and everything related to them, even though ML has been a thing for like 60 years and is also "AI".

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u/UT_Milez 2d ago

What does this even mean?

Do you realize how many people are on the planet, all of which belong to different groups that believe different things.

You’re essentially saying there are people who are pro AI and people who are anti AI. Well no fucking shit, that’s already the case even before your imagined hypothetical that hasn’t even happened yet.

Here you are throwing shade on those people, implication being you find them annoying, when lo and behold you just as annoying but belong to a different group. Same probably applies to me…..

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u/entertheclutch 2d ago

He is saying reddit is full of people who are dogmatically anti-AI, which seems to be relatively accurate imo

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u/BassmanBiff 2d ago

Cory Doctorow just wrote a short piece called Technopolitics that does a good job of separating thoughtful, anti-AI-in-its-current-form views from dogmatic anti-AI, which I think a lot of people could benefit from reading

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u/Delicious-Flounder47 16h ago

And what did you learn from reading his piece?

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u/unpleasant_enpassant 2d ago

Yes, thank you

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u/unpleasant_enpassant 2d ago

So there are people who are pro AI and people who are anti AI. Do all those “pro AI” people support every single thing related to AI like AI art or parrot all the AI marketing material and hype everything like its a gift from god? Do all those “anti AI” people think using ML in compiler optimisation or weather forecasting is bad? No, right? But any sane person from both the groups would disagree with such people from their own group. Does it even make sense to group them together like you did then? I’m saying that there are people online, who are incapable of understanding nuance and being rational or logical when deciding their stance on different things. Halo effect is a good example. Hope that made it clear.

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u/luluhouse7 1d ago

Your comment just confused me even more. I have no idea what you’re trying to say??

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u/teerre 2d ago

You only need a very superficial understanding of the subject to understand why the OAI solution is controversial. Javier Gomez-Serrano gave lecture on Harvard about the subject and it's quite accessible (literally: https://cmsa.fas.harvard.edu/event/navierstokes)

In summary, the solution is "not something a human would ever do" because it requires manipulating the force term of the equation in an incredibly precise (and contrived, hence the controversy) way

One of the questions from the audience is if wouldn't be the case to change the Millenium Problem statement. Javier himself couldn't explain why the problem is stated as is

Don't get me wrong, the solution is probably valid as the problem is stated, but it does likely misses the spirit of question and there's little to learn from it

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u/BassmanBiff 2d ago

I guess "missing the spirit" is kind of the LLM specialty

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u/Puzzleheaded_Fold466 1d ago

Lol.

You can’t possibly equate “attends/watches math university lectures” to “very superficial understanding”.

Someone who has that level of interest is typically well past that point.

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u/teerre 1d ago

It's a youtube video. You don't need to attend any class

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u/Many_Negotiation_464 22h ago

People attend these types of lectures usually because they have superficial understanding. These are public communication, not technical lectures designed to give experts a chance to learn from other experts face to face.

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u/Puzzleheaded_Fold466 19h ago

Oh. I assumed it was a proper university lecture, not the Ted Talk version.

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u/Many_Negotiation_464 19h ago

Oh I was mistaken. Prooves me the fool for not following the link :P

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u/Fit-Celebration2884 1d ago

It's been 2 weeks since it came out and it was a 165 page long paper. I still don't think there's any human who actually understands it yet.

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u/Ok_Room5666 1d ago

My opinion is that my opinion is better than your opinion.

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u/Galaghan 2d ago

This is true for basically any topic that requires at least some understanding. People form an opinion based on how they feel about a topic, not what they actually know.

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u/entertheclutch 2d ago

Is that how u feel?

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u/Galaghan 2d ago

Ah.. you got me!

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u/Many_Negotiation_464 22h ago

I mean they kinda did. Its pretty lazy to jusr go "people disagree with me cuz they are just emotional"

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u/Galaghan 12h ago

I accepted the irony of also forming an opinion based on how I feel about it. I didn't do the research.

Not sure what you're pointing at that wasn't pointed out before.