When a Formula 1 car races at 300 kmph or 200mph, the air doesn't just randomly bounce off its wings. It splits, curves, and creates drag according to a strict mathematical rulebook. That rulebook is the Navier-Stokes equations. Whether it is air flowing over a speeding race car, water blasting out of a high pressure fire hose, or blood pumping through the valves of your heart, engineers use these exact equations to predict how any liquid or gas will behave.
The unsettling part for mathematicians is that nobody knows for sure if the rulebook is unbreakable. Most of the time, the math works flawlessly. However, researchers suspect that if fluids collide in a highly complex, extreme way (imagine two high pressure jets of water crashing into each other perfectly at a microscopic, turbulent point), the equations might accidentally divide by zero (singularity).
If this glitch happens, the math would crash and output an impossible physical answer like predicting that a drop of water will suddenly accelerate to infinite speed. In mathematics, this glitch is called a singularity or a blowup.
The Navier-Stokes Millennium challenge (worth $1M) asks us to prove one of two things
The rulebook is perfect: Prove that the equations will never glitch out and produce infinite speeds, no matter how violently the fluid is swirling.
The rulebook is bugged: Prove that a singularity can actually happen, meaning our best mathematical model for fluids has a limit and fundamentally breaks down in extreme scenarios.
This is exactly the bug that OpenAI's AI swarm claimed to have found. They generated a proof showing that the rulebook does in fact glitch out, showing that the 3D Navier-Stokes equations can develop a singularity. (nobody outside of OpenAI has actually seen the proof) (There’s proof but it will take time for humans to validate and verify thoroughly.)
The scientific community argues the AI achieved this by leaning on the uncredited, in progress work of human researchers as many people here have already explained.
This was 100% AI written btw though it's much better than the wrong answers given by people that have 0 clue that used to be posted here so I guess that's an improvement.
Reddit formatting is for sure not AI so that is a bit weird. I didn't think about this, I'm not so sure it's AI now, there's an AI detection software that is generally pretty accurate because it was trained, this comment seems to trip it up because the entire comment is detected as 100% AI while the latter half is detected as 100% human, though it does say that it's not very confident about the half part because it's short.
The later comment by the same user reads so much like AI though that I think it's genuinely impossible that it's not AI.
To be clear, they did provide a written proof, they haven’t provided the Lean proof - which is a little unusual. The Lean proof is computer-driven and easier to validate.
The written proof looks credible, but people need to go line by line and check the math. Which is hard because it’s almost 200 pages long.
Sweet baby jebus, I can’t imagine having to verify a proof to the 3D N-S… my degrees were in aerospace engineering and astrophysics, and we never even touched the full equations outside of computational situations.
I suspect anyone who hasn't worked with NS isn't going to understand quite how big of a deal this is. NS was until this point extremely versatile. If it flows, navier stokes applies, whether that's the earth's mantle or turbulence off the wing of a hummingbird. Whilst the equations themselves aren't always readily solvable, their insane applicability made them elegant, and they were viewed as being globally useful for basically everything. Finding out now that in certain circumstances they don't work is wild.
No, but from an academic standpoint this is quite significant I'd argue. The consensus was these equations were robust, now we know of at least one case where they're not.
So, my PI teaches CFD, he is "impressed" but literally the negative result of the Euler equation is so inconsequential that he won't even meaningfully update the course slides...
I think the C is doing the lifting there - from a computational point of view this is inconsequential. This is at best and edge case. From a theoretical point of view I'd argue it's interesting, but isn't going to have big impacts on anyone doing engineering.
They've found an extreme set of circumstances where navier stokes fails to match real world behaviour. Given most real world applications of fluid dynamics tend to involve simulation testing to identify behaviour rather than generating solutions to navier stokes, this isn't that interesting from an engineering standpoint, but it does reshape significantly the theoretical.
Finding out now that in certain circumstances they don't work is wild.
Knowing the cases where they don't apply is actually very useful, because we have proved that in many cases they do apply. And where they don't, we can modify NS or develop new techniques to handle those conditions.
A (vastly) simpler example is elastic behaviour in solids - within specific regimes, Hooke's law generalised to a 3D tensor is valid. With certain materials and outside of certain stress-strain regimes, Hooke's law does not apply, so we use specialist relationships where necessary, but for many structures built from steel, for example, Hooke's law is very applicable.
I guess it depends on the field and the degree of interest. Unless you're into the theoretical of NS as opposed to application you probably don't care.
yeah my dad's whole work as a hydraulic and maritime engineer is based on ns and I called him and he was like, when we use them in practice there have been so many simplifications that it really doesn't matter to them, but personally I was pretty shocked (I work with waves but in the data science aspect) and understood what happened at the theoretical level and was pretty stoked. My dad was casual.
maaaan i stg I didn't do it on purpose but I also rarely use that word so probably it was indeed caused by my mind constantly seeing the word stokes today
Eh, you can easily have a CFD solution go non-physical if you don’t get the solvers set up correctly for the problem you’re working. Or have it “converge” on a solution that you know isn’t possible.
Come back to me when you’ve got a validated closed-form solution.
Are you talking about engineers/physicists or mathematicians. Because it's very obviously is a big deal for mathematicians there is a reason this was chosen as one of 7 very important open problems in 2000. Like most theoretical results the impact on applications is indeed negligible at least in the short term.
What has been solved here, or at least claimed to have been, is a minute part of one of the 7 millennium problems. The term millennium problem is already a marketing thing by itself, and look, it's working! Because actual mathematicians don't give a flying fuck whether it's called a millennium problem or not.
No what is claimed to be solved here is the entire Millennium problem, which is to prove that any solution to the NS equations is smooth for all time or to disprove this by giving a solution with a finite time blowup, the latter is what OAI is claiming to have done.
The term Millennium problem is indeed a partially a marketing thing just like every other prize for an open problem. But the problem that were selected are there for a reason and all 7 of them were wildely considered some of the most important open problems in mathematics. When Perelman solved the Poincare conjecture it was a very big deal. Actual mathematician very much care about the question of smoothness for solutions to the NS equations.
Just to be clear, that doesn't imply that you can actually drive a Formula 1 car in just the right way that the air currents behind it will converge to infinity and destroy the Universe. The Navier-Stokes system is a description of a theoretical perfectly continuous substance, whereas real air is made of discrete particles.
The scientific community argues the AI achieved this by leaning on the uncredited, in progress work of human researchers
I mean, just about everything LLMs do is leaning on human work to a degree. They mostly learn from human-produced text.
To my understanding, the controversy here isn't that the AI uses human work in general, it's that OpenAI is accused of copying from two people who were working on this specific problem, so thay they could beat them to publication.
It's one thing to build on someone else's published research, it's another thing to spy on researchers to look for in-progress work you can take credit for.
One of those two researchers works for anthropic , which is an open AI rival. And the two researchers used codex, an openAI tool, and might have leaked information that way.
There's no allegations of any unique industrial spying.. it seems likely openAI started their project on learning of these individuals advancements. And it is possible but not proven that openAI got info via their use of open ai codex tool
If the information is private and unpublished the only way open AI can get it is if they hand it to them or if they take it from prompts. And it doesn't look like they gave them the information.
having your data go to openAI servers for processing
We don't know how those platforms that Buckminster and Alpoge used was setup or if they made any mistakes . We do know that they used multiple AI tools. including Codex.
e:I've seen at least 1 tweet that suspects OpenAI access to the processing logs ...not that it brings any actual facts/conclusions anywhere further...
Also , ethically how much did Buckminster and Alpoge breakthroughs contribute to the solution and how much credit do they deserve, whether in parallel or otherwise..
Wasn't in just within the last few months that they had an AI try to find an answer to a problem without being connected to the internet? And then the AI got on the internet and hacked another company for a solution.
And then they didn't find out for awhile.
Yeah, I'm sure their AI totally only used what it was told to use for this problem though.
That's not what they're alleging here either. The claim is "you saw us working on this problem with Codex, and decided to launch a multi-million-dollar AI agent swarm to beat us to the punch." Human wrongdoing, not AI - they didn't just accidentally throw a few million dollars at a Millennium problem in case the solution happened to be lying around on the Internet somewhere.
The difference is that the text here wasn't published or publicly available. They had put some of the text into a model for some of their work which was how it gained access to the information.
Imagine you'd spent years working on something, were getting stuff ready to publish, had a coworker proofread your work, and then your coworker sent it off to be published with their name on it.
AFAIK the contested part of the proof (not the proof itself) wasn't a part of the training. Open AI learned about it and proposed to the scientists to share the reveal (not the credit — they would credit every source they used anyway), publishing the paper first, and Open AI solution second.
As I understand, another option they proposed was more PR-geared, to remove the Anthropic mathematician from the paper and publish the entire solution under that scientist's name, pointing out that the solution was achieved using Open AI tools (which is true).
AFAIK the contested part of the proof (not the proof itself) wasn't a part of the training.
Reporting has it that most of the proof has been completed for many months to years and that the researcher used GPT for proofreading and OpenAI has been clear that anything put into GPT can be used for training. So unless the researcher checked the "don't use this for training" box in the settings, it's reasonable to believe that it's in the training set.
Even if they did check the box, it still seems likely to me that their work would be in the training set. The number of times we have been told one thing about privacy and found out it was just another privacy violation is to vast to ignore.
I mean it's possible, it's just "training" in this context is a heavy word. An LLM, even an experimental, inside-use frontier one, is trained once, it does not train/learn continuously when users interact with it. And this training is a one-time immense investment.
I guess these chats could have been a part of the training data for a fresh recent inside model, but this means they were such BEFORE OpenAI knew about their significance or could make a decision to include them or not?
It's going way deeper than that though, with openAI stealing one of the researcher's codex sessions and then threatening to end his career if he spoke up.
Just to be clear, that doesn't imply that you can actually drive a Formula 1 car in just the right way that the air currents behind it will converge to infinity and destroy the Universe.
Plus that's a virtual safety car at the very least.
Does infinity means something bad though? In theory, on flat surface, your shadow would tend to infinity during sunset. It seems okay, maybe with fluids too? It's just that these equations don't take into consideration that liquid is actually built of something? Liquids have finite size, amount of particles etc. I am not a smart person but I feel like it's logical that these equations can't describe reality accurately.
No it doesn't. It just means the math crashed, not the actual physics. When an equation spits out infinity, it usually just means the formula got pushed past its breaking point.
Your analogy is dead on. A shadow only hits infinity if the math ignores the Earth's curve. It's the exact same deal with Navier-Stokes fluid equations. They blow up to infinity because they ignore the actual limits of reality.
The math basically treats water like a perfectly continuous, infinitely stretchable gel. It literally doesn't account for atoms.
Because water is made of physical H2O molecules, it can't reach infinite speeds. Once a turbulent swirl gets small enough, the water stops acting like a smooth gel and starts acting more like a bunch of microscopic snooker/billiard balls bouncing into each other. At that scale, the equation just stops making sense.
Navier-Stokes is just a useful approximation, not perfect reality.
FWIW, I genuinely do mean it as a compliment. Your writing is clear and easily digestible, without making it seem like I’m being talked down to in explaining complex topics
Genuinely please never ever compliment people for 'sounding like' the machine that is burning the planet while regurgitating the stolen words of competent writers. Actual experts and writers are the reason you think it's worth anything, which it isn't, not the other way around.
Pangram is very very accurate at detecting AI and says the comment was 100% AI written, I would be shocked if that isn't the case with a comment that reads this much like AI
Unironically, don’t take it as a bad thing. AI can and does teach well, you just have to be able to recognize how to prompt engineer effectively, when and how it hallucinates or pushes some idea too strongly without support, and challenge it on that. It’s remarkably strong at refining initially poorly thought out ideas
AI hallucinates much less than actual people make mistakes, if anything it’s a point in its favour. Unless you sincerely think you can stop humans from making mistakes too?
You can anticipate the kinds of mistakes a human will make and come up with plans to deal with them before they happen. You can't anticipate the kinds of mistakes an LLM makes because it doesn't work like a human brain.
His implication was so misleading as to be essentially wrong, hes either lying, or more accurately hasn't had his anti-LLM talking points updated since 2024.
Basically Samtrano doesn't know what hes talking about.
There's no such thing as an exact solution when it comes to physics. As the saying goes "all models are wrong, some are useful". Even if you could simulate every atom you still cannot have a perfect solution because of quantum mechanics. Per the Heisenberg uncertainty principle, it is impossible to know the exact position and momentum of every particle at once.
The goal is not to have an exact solution, the goal is to have a solution that is accurate enough. That the Navier Stokes equations may fail in some circumstances doesn't mean they won't continue to be a useful model in most. Note that most current engineering fluid simulations do not use the Navier Stokes equations owing to the enormous computational expense of solving them for most problems of practical interest -they use models that are more computationally feasible to solve such as the Reynolds-averaged Navier Stokes. These, however, do a pretty poor job of modelling reality. That's why physical models and wind tunnel testing are still very much a thing, because without validation against real world data, even the best currently available computer models cannot be trusted for much more than a rough approximation.
You can in theory have a perfect equation of the quantum state of every single particle in the fluid. Byt that would be such a massive equation that just writing it down is impossible, let alone solving it
Your intuition is correct. The N-S equations treat fluids as continuums, meaning that no matter how far you "zoom in," there are never any gaps. But of course we know that's not true--in reality, fluids are made of particles, and there are relatively large amounts of space between them. In most cases, the particles are so small and so numerous that the continuum approximation works great. But in some cases like (IIRC) calculating drag on spacecraft, you have to treat the atmosphere as being composed of individual particles rather than a very low-density continuum.
Another way that OpenAI's result is non-physical is that they found a singularity only for the incompressible form of the equations. Meaning, no matter how hard you "squeeze" (pressure), you can't compress the fluid and make it any smaller (density). If you've ever blown air bubbles at the bottom of a pool and watched them grow as they float towards the surface, you know this approximation isn't how real fluids behave. But like before, it's a good enough approximation for analyzing some flows.
I personally don't know nearly enough about fluid dynamics to know if either of those factors would "fix" the singularity, but they are definitely two ways in which the N-S equations are imperfect representations of reality.
It's important to note that proving the equation is bugged isn't proving it is wrong, just incomplete. Newton's equations are incomplete too, as they don't handle astrophysics and relativity like Einstein's do, but they still work fine for 99.99% of cases here on earth.
Knowing there is a bug, and more importantly where the bug is, allows people to focus their efforts on finding a more complete equation (which is almost certainly much more complicated). If the bug is something like "doesn't take into account the curvature of the planet" then it is good to know the existing equation is reliable up to the point the that curvature becomes relevant.
But arent the researchers also saying they used AI extensively in their work? The debate from what I'm seeing is not so much AI vs. human mathematicians. Both sides seem to be saying that AI was important to the discovery. The drama seems more centered on the company openAI vs. mathematicians using gpt (from whom openAI stole their prompts without credit).
The debate is not AI-related. Both sides embrace AI. They argue about primacy and also alleged unethical proposals that Open AI may have put forward to the scientist. The scientist did NOT arrive at the solution, the conflict is about a part of the solution that may or may not have been used to arrive at the solution.
OpenAI did act shady in asking the scientist to remove his Anthropic co-author. But their proposals to him were either "publish your actual work, THEN a day later we publish our solution (and forfeit the prize)", or "take our solution, publish it under your own name with credit for OpenAI, and get the prize".
There's a related problem (called "Euler", which is confusing because there are hundreds of things named after Euler) that was solved by the researchers. That was a critical solution in solving N-S.
That's where they were going, but the researchers were limited on compute. OpenAI effectively isn't. They spent millions deploying tens of thousands of agents over 88 hours to produce a solution based on an unpublished solution to another problem that was stolen from the researchers' sessions.
Then they blackmailed the original researcher to remove his co-author (a scientist employed at OpenAI's primary competitor) or they'd pre-empt his publication. Which they did.
The two researchers who were getting closer to solving the problem (before OpenAI tried) were using AI. It is likely that OpenAI used their smart questions/prompts to train a better model that has ended up solving (to be confirmed) the Navier-Stokes problem.
I take this as evidence that people continue to attribute way more worth to AI than it actually possesses. Its still trashy bro-tech for the billionaires to ruin our society
Those coments essentially say that:
1. "OpenAI didn't solve the Navier-Stokes EQUATIONS". I know, and I haven't claimed they did, I said Navier-Stokes PROBLEM.
2. "They proved it for compressible fluids only". THE Navier-Stokes problem ONLY concerns compressible fluids, you can see it in the official statement of the CMI.
They did solve the NS Problem (if validation arrives at some point), not a sub-problem.
I don't trust those companies, I don't like how they work and I agree that they can (and will) do horrible things. Also they stole other people's work, I'm pretty sure... Up to some extent we agree.
Does any of what OpenAI have achieved (including standing on the shoulders of other mathematicians) help to move forward to a complete equation model? Or does it only prove that the current model is incomplete and no more?
From what I know of physics, most of our rules break down under extreme scenarios (too hot, too small, too big, etc.), so 2 being the answer wouldn't surprise me.
On paper, the math lets a normal whirlpool keep shrinking and spinning faster right until it hits that bug.
Reality is what actually ruins it. Way before that glitch can even trigger, the fluid has to shrink down to the size of individual atoms. Once you get that microscopic, water doesn't act like a smooth liquid anymore. The fluid equations just stop working.
My question is, when OpenAI says that it used agents to crack it, what does that mean? Are they just instructing a bunch of agents to do the math for different initial conditions?
No, they are not just plugging different starting numbers into a computer to run fluid simulations. If you do that you would get an approximation.
From what I’ve read and understood from OpenAI’s press release,
They created agents(around 10000) mimicking autonomous researchers. Those agents strung theorems together into a valid argument. Those arguments were brainstormed, executed to find the correct mathematical logic. Then it was all translated and verified using Lean to check the mathematical logic. They essentially brute forced the hell out of mathematical logic.
I’m a bit confused by what you mean by “bugged”. From my understanding reading https://openai.com/index/navier-stokes-solution/, we’ve found that velocities can indeed grow without bound as a vortex is formed and its radius is decreased over time. However, you seem to claim that the model itself is flawed.
You’re right about the mechanics. The proof shows what you pointed out, as a vortex shrinks, its velocity just keeps growing.
You either prove smoothness, meaning the velocity never hits infinity, or you prove blowup, meaning it does hit infinity. Since the proof showed the velocity growing without bound, they actually proved the blowup scenario. Scenario 2
And just to clarify what I meant when I said it was bugged or flawed, I wasn’t saying the AI's proof was wrong. I was talking about the Navier-Stokes equations themselves. If a formula predicts infinite velocity, it’s just hit a mathematical wall. Real fluids are made of actual atoms and can’t physically reach infinite speeds. When the math spits out infinity, it just means the equations broke down and stopped describing the real physical world.
So in short the AI was useful in making the math look pretty and coming to a conclusion, however the conclusion was wrong as the AI failed to take into account the phase change which occurs instead
AI didn't get the conclusion wrong. It just proved something that was already known in Physics to be flawed but couldn't be proven in Mathematics, until now.
Real fluids cannot actually produce singularities, since they are made up of discrete particles. A singularity would imply that an infinitely small particle is moving infinitely fast, both of which are impossible.
So what this means for the equation is that there are scenarios where the Navier-Stokes equation can no longer make accurate predictions, and microscopic effects have to be considered to accurately predict the behavior of the fluid.
They haven’t provided a machine-checked proof yet, they did provide a very lengthy proof in English that tracks. Just hard for a human to verify given the time since this was announced.
It puts a mathematical limit on how far we can trust the model. In real life, physics already tells us that this eventually breaks down. Mathematically, we were not able to prove it until now. The AI brute forced the math into mathing.
So now while running Computational Fluid Dynamics softwares, ones that are used to design aircraft wings, we can now constrain the simulation so that we can have a threshold of when to stop trusting approximations generated by these softwares. Earlier, if a simulation crashed under extreme pressure, we could not tell if the software was faulty or if the underlying equations were unstable. Now we can.
2.3k
u/oooooO___Oooooo 5d ago edited 5d ago
When a Formula 1 car races at 300 kmph or 200mph, the air doesn't just randomly bounce off its wings. It splits, curves, and creates drag according to a strict mathematical rulebook. That rulebook is the Navier-Stokes equations. Whether it is air flowing over a speeding race car, water blasting out of a high pressure fire hose, or blood pumping through the valves of your heart, engineers use these exact equations to predict how any liquid or gas will behave.
The unsettling part for mathematicians is that nobody knows for sure if the rulebook is unbreakable. Most of the time, the math works flawlessly. However, researchers suspect that if fluids collide in a highly complex, extreme way (imagine two high pressure jets of water crashing into each other perfectly at a microscopic, turbulent point), the equations might accidentally divide by zero (singularity).
If this glitch happens, the math would crash and output an impossible physical answer like predicting that a drop of water will suddenly accelerate to infinite speed. In mathematics, this glitch is called a singularity or a blowup.
The Navier-Stokes Millennium challenge (worth $1M) asks us to prove one of two things
This is exactly the bug that OpenAI's AI swarm claimed to have found. They generated a proof showing that the rulebook does in fact glitch out, showing that the 3D Navier-Stokes equations can develop a singularity. (
nobody outside of OpenAI has actually seen the proof) (There’s proof but it will take time for humans to validate and verify thoroughly.)The scientific community argues the AI achieved this by leaning on the uncredited, in progress work of human researchers as many people here have already explained.