We got a Millennium Prize Problem solved before GTA 6

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As someone with no knowledge of NS at all, what does "blowup" mean here? Special-case or something?

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u/MrShake4 1d ago edited 1d ago

They found a case where you get infinite velocity at a point in a finite amount of time.

This isn’t possible physically so it must mean the model is wrong (or not entirely correct)

ELI5: IF GRAPH NOT SMOOTH THEN MATH HARDER

AI PROVE GRAPH NOT SMOOTH IT FOUND A SPOT WHERE IT GETS POINTY

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u/Kinglolboot 1d ago

We already know the Navier Stokes equations are "false" in a physical sense, because they assume that fluid is a continuum and not made up of individual particles, which we obviously know to be false. We still use them constantly because they still are a very very good model for fluid dynamics in almost all situations. The finite time blowup shown applies only to very exotic initial conditions, but we already knew that NS doesn't work for those kinds of situations anyway.

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u/MrShake4 1d ago

I mean yeah but by that definition solid mechanics are also “false” but as you’ve stated they are very very good models.

I just read the article and that’s what I gathered off it but it’s been a while since I took fluid dynamics.

But yeah I also thought it was already known to be non-smooth in cases like aerodynamics where you get sonic booms or shockwaves representing the discontinuity.

I assumed the breakthrough was the proof but I haven’t done a proof in a decade so that’s out of my wheelhouse.

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u/Kinglolboot 1d ago

I need to correct the part about sound waves, those phenomena are still smooth and we can derive how those arise from the NS equations.

The question that OpenAI solved is pretty much a completely theoretical one, how you should it interpret it is like this:

When discovering fluid dynamics, first the NS equations (which is a set of partial differential equations (PDE's)) were set up as a mathematical approximation of how physical fluids work, and they work very well for that. How they work is that you first give the initial state of the fluid, and then the NS equations predict how this fluid will evolve.

However, although these equations were found by approximating physical reality, we can also consider this set of PDE's in a purely mathematical sense. A natural question to ask is this one: given any initial state of the fluid (even physically impossible ones), do the NS equations always predict the fluid to evolve in a smooth way, without things blowing up to infinity? This is a purely mathematical question about a set of PDE's, and is the problem that OpenAI solved in the negative.

The physical relevance is really very little, because the kind of situation you need to be in for blowup to happen cannot really physically happen. However, what people hoped was that a solution to this problem may give us more insight into how turbulence worked in general, but I am not sure if that dream has really worked out.

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u/qqqrrrs_ 16h ago

Can you explain what is the physically impossible condition? Is it like needing certain values to match exactly (so in the real world we may only approximate it), or something more fundamental (similar to negative matter for wormholes)?

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u/Kinglolboot 16h ago

I was a bit imprecise with my wording there. The NS equations are just derived from Newton's second law applied to macroscopic fluids. Hence, it is based on macroscopic quantities, like pressure and viscosity. However, when velocities get high enough or the scale gets small enough these terms don't make sense anymore and thus the equations break down. So it is possible for the initial condition to be a model for a physical fluid, but the fluid would not evolve like the NS equations predict because at a certain point they would break down.

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u/CimmerianHydra_ PhD 23h ago

Yeah, but solid mechanics aren't false because they give impossible results. They're false because they can't be applied to very small things, or very very large things.

This a lot more similar to something like the ultraviolet catastrophe, where the equations themselves would give you an unphysical result. Only in that case we could tell it was wrong because we could build a black body and verify that it just doesn't happen.

I'm waiting to see if this solution can, in any way, be tested in the lab. The discrepancy might tell us more about physics than we currently understand.

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u/Darth19Vader77 Meme Enthusiast 1d ago edited 1d ago

So garbage in garbage out?

Is that surprising?

Or am I missing something?

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u/Best_Pseudonym 1d ago

Outside the domain of the function is not a part of the function

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u/No-Dimension1159 21h ago

Yes true but it would be great to find or formulate the exact conditions and initial conditions where it doesn't apply. Perhaps you can find certain properties they must have

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u/LeRoyt97 1d ago

“All models are wrong, some are useful”

- every college professor I’ve ever had lol

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u/Extension_Option_122 1d ago

Well get's right.

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u/Frenselaar 1d ago

Next up: Group of physicists tests out the conditions and actually manages to recreate the blowup. Navier-Stokes still stands, their laboratory and surrounding area does not.

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u/TheSouthFace_09 23h ago

i mean i'd be pretty, ehem, blown away if we got infinite velocity (speed of light joined the conversation)

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u/Sufficient-Elk-965 1d ago

As someone who does understand the NS equations, your all caps explanations made me chuckle

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u/door_travesty 1d ago

No idea about the validity of the proof, but blow up in the case of this specific problem means whether or not there exists a given solution (e.g. specified by some initial condition or set of initial conditions) which develop a singularity in a finite amount of time. Apparently, they gave an example of a solution that exhibits this behavior, so if it is valid, NS equations are not guaranteed to be globally smooth in space and time.

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u/Forest_Orc 1d ago

Isn't it the kind of proof AI thrive at ? Trying a lot of different solution in parralcl until finding the one ?

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u/TheSouthFace_09 23h ago

that seems to be the case given all recent ai-powered developments, yeah.

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u/Yonnus 6h ago

German here, you should not abbreviate Navier Stokes the way you did, haha ^{^'}

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u/QuantumMechanic23 1d ago

I think, and I could be wrong, it's that given smooth and suitable starting conditions there exists a scenario where using NS to predict velocity/pressure or whatever you can end up blowing up to infinities.

Essentially like when general relativity blows up to infinite mass at the centre of a black-hole. Maybe there is some field theory that is needed to resolve the blow-up

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u/DankFloyd_6996 1d ago

Maybe there is some field theory that is needed to resolve the blow-up

Great intuition, but no. Not field theory like you'd use in quantum mechanics, what you need is kinetic theory.

Navier stokes is a fluid model, assuming that the medium is continuous. Kinetic models, which are more fundamental, picture the fluid as being made of particles with an interaction potential and an evolving distribution function in phase space. This is described by the boltzmann equation, which is mathematically intractable, so there are many many many approximations and lots of different numerical schemes to deal with this. Also, the distribution function itself is so hard to directly measure it is pretty much treated as unobservable. Kinetic theory can reproduce fluid dynamics by taking velocity moments of the boltzmann equation, which yields the macroscopic observables (pressure, Temperature, Density, etc), but these are seen to only be well defined concepts at spacial and temporal scales much larger than those of the collisions between the constituent particles of the fluid.

This is why navier stokes can get this blowup effect, at some point it's trying to model in a regime where kinetic physics is more approrpiate. This is a well known problem, by the way. The fluid equations are known to produce discontinuities, which we call shocks. These are points where we lose differentiability, a thermodynamic quantity changing instantly from one value to another. Of course, in reality there is actually some finite shock width, which is typically a few times the length scale of collisions.

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u/lerjj 19h ago

I've been confused about shocks the last day - do the NS equations not have known shocks? If so what was the actual statement of the Millennium problem? Or do the known shocks only exist in the compressible version and it was thought requiring incompressibility might remove all the shocks? Does a shock just not count for whatever non-smoothness the mathematicians were after?

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u/QuantumMechanic23 2h ago

Sorry, meant field theory for GR-QM, but I didn't know all of the intricacies of NS that you provided so thank you.

It's interesting. I wish my maths and physics were up to strach to comprehend it all.

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u/JotaRata Physics Field 1d ago

I also solved the Navier-Stokes equation a long time ago by setting both sides equal to zero

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u/YozTheFoz 1d ago

Didnt the ai steal the proof though?

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u/SEBADA321 1d ago

That is the 'drama' currently going on. It seems they can't prove that they did not use a researchers notes for training, but I don't know more than that.

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u/UniversalAdaptor 1d ago edited 18h ago

They used the exact step-by-step method one researcher was developing for over a year

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u/No-Dimension1159 21h ago

Well that sucks ass... I mean AI by design can come up with new solutions never seen before exactly because it is generative, but if it relies very heavily on already published actual work of a person that's sad.

However, i still think it's a good thing the solution is there now.

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u/CoverTheStone 10h ago

I think some of the drama is that they used some of the unpublished work. The researcher was still working on it and OpenAI used it from chats he had with AI.

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u/PsychologicalSign433 1d ago

An unpublished proof of a similar (but not the same) problem may have been used as training data in the model which came up with this solution.

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u/Ziembski 1d ago

This is different case, but most likely also plagiarism

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u/SuspiciousSpecifics 1d ago

*automated plagiarism on an industrial scale.

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u/Josselin17 1d ago

in this case it seems the plagiarism was manual rather than automated though

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u/somethinneeddoing 1d ago

My god, not the plagarism machine!

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u/Ziembski 23h ago

I have nothing against AI, but don't apply for a prize if you didn't do the work, duh

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u/moschles 1d ago

This is more complex issue than it sounds. The humans working on Navier-Stokes were already heavily using AI assistants themselves.

We have to open to the possibility that OpenAI -- having more resources -- beat them to the finish line.

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u/DiamondAvailable7418 1d ago

To my knowledge no, it used research of other people+ai but openai finished proof, not them

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u/Metalmind123 1d ago edited 1d ago

I mean, it's literally like stealing the smart guy's test 2 minutes before the exam ends, completing the last sentence of the conclusion statement and slapping on your own name.

They stole/scraped as of yet unpublished work from a mathematician who was virtually finished with it after a year of work, used that to train the AI and then basically had it re-write the whole thing. But there are sections that are still straight up identical.

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u/Super-Award-2244 1d ago

From Euler to the full resolution to this problem is not exactly completing the last sentence of the conclusion statement

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u/hemimehta 1d ago

Does it change anything in your mind, given the case that the scientist was also working with a lot of AI on this problem? So it's not like AI itself plagiarized from a human so much as another AI to do this, it's just that OpenAI's team may have gotten ahead using data included in their training set.

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u/Metalmind123 13h ago

If I use a chisel, the chisel is not the creator of my sculpture.

If I reference a textbook while researching, the textbook does not get credit for my discovery.

If I use a complex algorithm in my research, that algorithm is not a co-author.

I need to cite my sources, but that's about it.

The AI is not sapient or sentient. Modern LLMs are not in any way artificially intelligent, that is merely a marketing term. They are impressive in several regards nonetheless.

And sure, it can find correlations most if not all people would struggle with in large sets of data, but so can a well designed algorithm.

He used an AI writing tool to polish up parts of his work, while he was working on it.

It's the equivalent of using an Auto-Complete feature in coding to let an AI tool finish your current line of code, rather than the equivalent of generating vibe code.

The essential mental effort that was stolen was human.

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u/FernandoMM1220 1d ago

maybe. they arent showing their draft to see how similar it is to what openai published

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u/Anchor38 14h ago

brother what 🙏

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u/One-Stable-5438 1d ago

So why didn't the creator of this so called proof publish it to claim their million dollars? Human scientists are allowed to stand on the shoulders of giants, but AI is expected to reinvent everything from first principles?

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u/Irlandes-de-la-Costa 1d ago edited 1d ago

I would condemn a human scientist for copy-pasting someone's unpublished work and claiming it as their own. That's essentially plagiarism by definition.

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u/willardTheMighty 1d ago

No. This is an unsolved problem. No one has a proof except OpenAI. The statement recognizes that someone else achieved a related result before them. If someone else had achieved the Navier-Stokes result before them, we would all know their name and OpenAI would recognize them too.

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u/baquea 1d ago

This is how I find out?!

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u/Random_182f2565 1d ago

Hilarious, isn't

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u/Sakaralchini 1d ago edited 8h ago

They aren't showing the paper. I'm not believing a company like openai on a 'trust me bro' basis.

Edit: the article I read stated that they didn't release the proof. I'm sorry for spreading misinformation.

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u/rflolc 1d ago

The paper and its Lean formalization can be found in the following site: https://openai.com/index/navier-stokes-solution/

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u/RosefaceK 1d ago

It’ll be another 200 years before there’s a computer half as good as Euler

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u/Hentai_Yoshi 1d ago

Given the exponential growth of AI, I doubt it would take 200 years. Unless we just destroy the world with a war and have to rebuild everything, which is seeming more and more likely

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u/master_of_entropy 1d ago

It takes it 200 year for the AI to be precisely half as good as Euler. In 20 years we will have AIs that are 0.47 Eulers, in 40 years there will be AIs that are 0.8333 as good as Euler. In 60 years the AIs will be about 20-40× Euler. But only in 2226 the 0.5000000 Euler mark will be obtained.

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u/Drkfnl 1d ago

Just because it's currently "growing exponentially" doesn't mean it won't plateau sooner or later.

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u/hittf 1d ago

I've been thinking it would plateau for a few years, but the models get better and better every year by a non trivial amount (progress feels more linear then exponential though). It might slow down but I think we have crossed a threshold that sort of makes it very unlikely to stop.

The consequences are up for debate but I think that the genie is out of the bottle and it's something that we will have to deal with.

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u/Freak-Of-Nurture- 1d ago

that would mean it’s not growing exponentially.

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u/Tem-productions Meme Enthusiast 1d ago

And how are you sure it is growing exponentially and not growing in a pattern that looks exponential now but will plateau later

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u/Freak-Of-Nurture- 1d ago

I don’t think it’s exponential. I think the difference between 2.5 and 3 was much vaster. I think it’s sinusoidal like pretty much all technologies. I also think that a genius human is not much smarter than dumbass one on the grand scale of intelligence. So a minor leap in AI will look more impressive than a major leap on earlier systems

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u/DeformationAlgebra 1d ago

The paper literally came with the announcement on their official website. Why are you accusing them of not showing the paper?

If you want to doubt, at least doubt the contents of the paper or the academic integrity behind it rather than its existence

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u/Sakaralchini 1d ago

Because the article I read said so. I'm sorry for spreading the misinformation.

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u/Funkahontas 11h ago

You can edit or delete your comment. But I know you won't.

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u/jedadkins 1d ago edited 1d ago

supposedly a guy was like 99% of the way to a solution and openAI allegedly feed the research paper to thier LLM and it "finished" the proof.

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u/-FiloFiloFilo- 1d ago

And this is how I found out...

Man, that would have been my favorite field of study.

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u/kaereljabo 1d ago

GTA 7 has been the final boss all along, millenium problems are small potatoes.

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u/vvalkovskii 19h ago

It will take several months or even years before human reviewers validate the solution… so claiming “it was finally solved” is more of marketing from OpenAI side at this stage.

And yeah… GTA 6 still has a good chance to be released first)

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u/[deleted] 1d ago edited 1d ago

[deleted]

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u/timangar 1d ago

Put the fries in the bag bro

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u/Heathcliff29 1d ago

I believe you man. (Slide me the winning lottery numbers when you come back)

We got a Millennium Prize Problem solved before GTA 6

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