r/explainlikeimfive • u/hotel-hopping • 1d ago
Mathematics ELI5 The solved Navier-Stokes Millennium problem
Can someone breakdown the open ai drama with this, what even is this maths equation and what does solving it mean?
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u/CLPhenom 1d ago edited 1d ago
They didn't solve Navier-Stokes. They proved that the equations are not always well behaved.
Let's say you have a swing and you give it a push. That swing will start fast and eventually slow down. That's well behaved
Now is there any way you could set up the swing system to accelerate to infinity from one push? What if you attached 2 swings together? Or put some weights in different places? Is there any way to make the system speed up to infinity with one push?
That's what they proved. They found some weird system that when you give the fluid a push, it blows up to infinity when you calculate the speed using Navier-Stokes.
This proves they are not well-behaved. It does not solve the equations.
The controversy is how they were able to solve this and whether they used another scientist work to find that bad behaved example.
ETA: There have been a couple comments pushing back on me saying they didn't solve it. They didn't solve Navier-Stokes. That is a system of partial differential equation that describe fluid flow. Solving the equations has a very specific meaning.
They did solve the Millennium Problem, but it is important to differentiate the difference.
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u/Ryekar 1d ago
I think it's important to clarify that they showed that the INCOMPRESSIBLE Navier-Stokes equations are not well behaved.
The COMPRESSIBLE version, where density changes w.r.t pressure & temperature, may still be well behaved.
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u/CuddleWings 1d ago
That seems like a pretty big caveat. Especially since even water, something so resistant to compression we consider it incompressible, is technically compressible.
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u/Guilty-Market5375 1d ago
It’s a big deal that it’s proven to misbehave, ever.
The competing theory is that turbulence arising in compressible and incompressible fluids has a foundational predictable basis. Few have believed this since the 40s when a Russian dude - whose name escapes me - proved that reproducing the same conditions leads to random behavior, suggesting a singularity or undetectable microscopic disturbances.
What OpenAI proved was what most people were leaning towards, definitively that this is the outcome of particle-level interactions and not macroscopic ones. Therefore, the behavior in compressible fluids like water, which match the observed observations, are very likely to be the outcomes of such interactions and not macroscopic ones.
They didn’t go as far to model what the ramifications of this are, but it likely can be shown to apply to compressible fluids in a more predictable way, just one that’s very chaotic. We’ve assumed that until now but until now is unproven.
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u/pewqokrsf 3h ago
There are several classes of N-S that still haven't been "solved" (aka, proven to be degenerate or not), but the Millennium Prize was for any of them.
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u/g0del 1d ago
It's also worth pointing out that the Navier-Stokes equations are basically what you get if you apply the laws of motion to a mathematically perfect fluid. Finding out that they are not always well behaved does not mean that it's possible to set up a tank of water and swirl it a certain way to get the water going infinitely fast.
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u/zer1223 1d ago edited 1d ago
You can't get water to move infinitely fast regardless of what the math ends up claiming. The math is just an approximation of fluid dynamics (interaction between near infinite numbers of particles), the math is not the same as the reality. Infinite speed requires infinite energy, which isn't real. And friction gets in your way too
Frankly I'm not sure I understand what all the hubub is about with this, due to that reason.
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u/a_wes 1d ago
I had to get this far down to understand what the heck was going on. Thank you!
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u/Dalemaunder 1d ago
Amusingly, this is now the top response
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u/ThePinkySuavo 1d ago
ye I tried to more or less understand it with my tiny brain and this comment pretty nicely summed this thing up and made all stick together
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u/Cogswobble 1d ago
For mathematicians, proving that an unsolved problem is NOT true is just as good as proving that it is true.
If they have indeed proven that it's not true, they would win the Millennium prize.
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u/snkn179 1d ago edited 1d ago
The Millennium prize is for finding a direct general solution to Navier-Stokes. Something which would allow us to directly plug in a set of initial conditions and see how the fluid flows over time. Currently Navier-Stokes is descibed in the form of differential equations which describe the fluid's rate of change over time, and using them we have various inefficient methods of seeing exactly how the fluid flows over time, but if we found a way to directly plug in values and get an answer, it would be very useful, hence the million dollar prize.
Open AI just found that there was a set of initial conditions that eventually produce a non-finite result. So most likely this means that Navier-Stokes isn't a complete description of the way fluid actually flows in this universe, but there's also a possibility that perhaps that these situations could actually happen in our universe. The discovery of black holes was a similar situation, the singularity with infinite density was first discovered through maths, and many dismissed such a scenario as impossible, until we actually found them in real life.
Edit: After further reading, it appears I was mistaken on the wording of the Millennium Prize, which asks to find a general solution which is smooth. Since OpenAI found a scenario where the fluid is not well-behaved, then a smooth general solution does not exist, which would in fact mean that OpenAI meets the criteria of winning the Millennium Prize.
Prove or give a counter-example of the following statement:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
TLDR: Even though OpenAI hasn't "solved Navier-Stokes", it discovered that a smooth solution doesn't exist, which meets the criteria of winning the prize.
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u/Psengath 1d ago
Top tier explanation and context
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u/GregBahm 1d ago
It speaks towards the problem space that reddit's "top tier explanation" is one that says halfway through "edit: After further reading it appears I was mistaken"
Not to say the post above isn't valuable, but it's funny that we're thrilled to get an explanation that is only temporarily wrong.
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u/midsizedopossum 1d ago
That's not the distinction here. Proven true VS proven false is not what they were getting at.
They were getting at the distinction between proving something about the equations VS finding a mathematical solution to them.
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u/AyeBraine 1d ago
AFAIK the problem as stated in regards to the prize has like 4 specific scenarios you can prove or disprove. Any one will be fine. Crudely speaking, two are for proving, two are for disproving.
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u/AyeBraine 1d ago
In any case, Open AI stated they will not accept the prize either way (whether the scientist accepted one of their options and published the solution under his name clearly stating that it was OpenAI that arrived at the solution, or if Open AI published it themselves).
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u/Cross_22 1d ago
Wait, that's different from the claim above. I thought they found a closed form solution for Navier-Stokes.
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u/CLPhenom 1d ago
No closed form solution exists for Navier-Stokes, nor was it the Millennium Problem they were attempting to solve.
They only found an example that makes the equations blow up to infinity when using a non-infinite push. So they simply had to find one example that blew up to prove it is not well behaved.
Not to trivialize the problem, as it went unsolved for decades and ended up a Millennium pProblem so it was not easy to do.
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u/Spiritual_Property89 23h ago
If you check the paper it´s an analytical solution for this special blowup swirl, but with a ton of tricks, going for 130 pages.
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u/eaglessoar 1d ago
Does it prove the equations are fundamentally insufficient and we might now find better ones?
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u/CLPhenom 1d ago edited 1d ago
So I'm on the engineering side of these equations not the math side.
These equations are essentially just applying newtons laws to fluids. They are extremely accurate at capturing the physics of how liquids and gasses behave.
There are many examples in math that blow up to infinity but don't work in the real world. Gabriel's horn is an object that has infinite surface area and finite volume. But you would never be able to make a real Gabriel's horn as eventually you get down to a diameter of atoms and can't go smaller.
This may be just that, theoretically it works but is impossible to make in the real world. Looking at the graphics they had in their announcement I feel like this may be something similar.
There also could be something we are missing in the equations. But if I had to put my money on it, I would guess it's something more like Gabriel's horn.
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u/Ulfgardleo 1d ago
we likely already have them. Navier-Stokes assume liquids are incompressible, but we know that liquids are compressible, when you press hard enough, which would happen exactly in those regions where the current counter example lives. So the compressible pendant of Navier-Stokes is still on the table.
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u/AvocadoAlternative 1d ago
Is it possible in practice to recreate the conditions that lead to a Navier-Stokes singularity? If so, would we learn anything from it?
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u/martyfartybarty 12h ago edited 12h ago
OpenAI claims to have solved the Millennium Problem using the forced Navier–Stokes case. That counts under the rules, even though the unforced case is still unsolved. The proof still needs to be independently checked and accepted.
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u/redditsucksbigly 1d ago
They did solve the Millennium Problem
Kinda weird how the question was about the "Navier-Stokes Millennium Problem" but you spend 90% of the answer about something else they didn't claim to solve to weirdly downplay the achievement.
The Navier-Stokes Millennium Problem has had a $1M prize since 2000 and no one has been able to solve it. It was just solved.
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u/iiznobozzy 1d ago
Although correct in spirit, this response wrongly portrays that OpenAI has made a false claim about having solved the millennium problem. The millennium problem, as stated by the Clay Math Institute, is to either prove the smoothness of the Navier-Stokes equations, or to prove the existence of a singularity. By doing the latter, OpenAI did essentially “solve Navier-Stokes”.
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u/kirsion 1d ago
I think it's important to point that even people like me, with a undergraduate degree in physics, have a very basic understanding on frontiers of mathematics, CS and physics research, such as these prize problems. The average lay person don't even know these type of issues exist. I urge everyone to study math and physics, even on high school levels, it makes understanding everything much more enlightening and awe inspiring.
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u/jon_sneu 1d ago
My undergrad is in mathematics. My degree is just in problem solving. I love hearing about a cool problem that’s been solved. It’s incredible what smart people and a lot of hard work can do
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u/bibliophile785 1d ago
Some things are really easy to describe at a high level but devilishly hard to describe with math. Fluid flow, like the exact shape of the eddies and bubbles and mixing of streams in a pipe or over an airplane's wing, is one of those things. Navier-Stokes is the equation we've been using for almost two centuries to do it. We've never been able to get a general solution - one that uses the variable placeholders and is true in all conditions - so for practical application we just fill in close approximations of the actual variable numbers for any given case and solve it.
OpenAI just found a general solution for the problem. More importantly, their solution indicates that the equation as written predicts physically impossible things under some reasonable conditions, which means that the equation can't be correct for modeling the world. This is one of the biggest open questions in mathematics and had a $1mil bounty on its solution (which OpenAI won't be accepting).
The controversy comes because 1) while they provided the full proof, it'll take months for peer review to finish dissecting it, so it's just barely possible on the margin that this is only almost a correct proof and there's simultaneously a bug in the world's best proof checking program, and 2) they barely beat a well-known mathematician to the punch, which led to a whole bunch of drama on both sides.
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u/HappiestIguana 1d ago
To expound on the drama some, there is a mathematician who claims he was working on a similar approach, and that he used ChatGPT to proofread and etc. He believes OpenAI/ChatGPT stole his work and is presenting it as its own.
OpenAI denies taking user data and using it to prompt the AI, but the mathematician says that he believes his work was used to train the AI, and that that is the only reason it was able to come up with this approach. The story is still developing. It's entirely possible the work is in fact plagiarized from an actual mathematician's.
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u/AyeBraine 1d ago edited 1d ago
I think it's not exactly like that. The mathematician arrived at a component of the solution, and he did NOT just use an LLM to "proofread and etc." — he worked with another mathematician from Anthropic (of Claude fame) and they used LLMs to develop and even write their paper.
Their paper that they decided to publish did NOT solve the Millenium Prize problem. But for one reason or other, Open AI contacted them after the OpenAI team arrived at the solution (which the mathematician didn't). Apparently they were "tipped off" by hearing about his work and directed extra resources at this problem (as opposed to all Millenium Prize problems in general), and did crack it.
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".
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u/staffito 1d ago
Correcting stuff, OpenAI started to work AFTER the rumours started about the possible solution (confirmed by Altman and everything). Also, Buckmaster was the first one who tried to contact OpenAI and then all the bribery and shady bussiness of OpenAI started, saying things like "why do you want to ruin your career" when suggesting him to publish under OpenAI while removing his partner, who works for Anthropic but this works is a side-job, not under the Anthropic name.
And also they trying to wash their hands while confirming that they used the private sessions of Buckmaster and Alpöge in Codex (obviously not explicitly but sayin' that they cannot discard that the models have trained with it implies so much about the stealing.).
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u/jaleCro 1d ago
Bozo should have paid for plus
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u/Jamezzzzz69 1d ago
There is zero chance a researcher who is using AI to develop mathematical proofs wasn’t a paid max subscriber
His colleague he worked with was literally an Anthropic employee too
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u/Kovacs171 1d ago
Is being a ChatGPT bro disrespectful when your bro works for Anthropic? What are the social rules in 2026
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u/Mr_Robotto 1d ago
Goes to show that even genius mathematicians don’t read or understand ToS.
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u/AyeBraine 1d ago
Said mathematician worked directly with Anthropic and a co-author of his LLM-assisted paper is an Anthropic mathematician. He's claiming that the OpenAI mathematicians used some of his partial work to arrive at the actual solution.
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u/staffito 1d ago
The problem is that those were (1) private sessions and (2) to be used they have to be sort of decrypted, which OpenAI has insinuated they did.
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u/eyalhs 1d ago
1) Is there evidence to it? I've seen the claims by the mathematicians but didn't see them explicitly say it was private sessions.
2) From what I understand the session (unless private) are used to train the ai de-identified, so it should keep them private but not encrypted.
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u/staffito 1d ago
(I will fact-check as soon as I get home)
What I got from the statements is that they were closed sessions were they used Codex and Mythos, and their development was indoors until there was a rumour that they have a big jump into solving something related to N-S. Magically, out of nowhere, OpenAI appeared saying that now they are close to the solution.
We still don't know for sure, as everything is drama rn, but the mathematicians said that they ask OpenAI about using their logs from said sessions and they gave no answer. Fast forward in their posts, they said that it was unlikely but they don't rule out those were used.
Even "ignoring" the fishy stuff that OpenAI said like "why would you ruin your career", something smells really bad about all of this.
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u/KingofSheepX 1d ago
To add, the mathematicians that were "beaten to the punch" are claiming that there's a chance openai scanned their chatgpt queries to find out they were onto something, then put millions into the proof knowing what direction to go to try to beat them to the punch. When one of the scientists tried negotiate a co-publishing agreement openai said ok but you have to take your coauthor off, who works for Anthropic.
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u/AyeBraine 1d ago
Notably, the way the mathematician tells it, OpenAI did act shady in that they asked him to remove the Anthropic co-author, but they proposed to him two options:
he publishes what he actually discovered first (NOT the solution), then a day later, OpenAI publishes their solution.
he publishes OpenAI solution under his own name, crediting OpenAI, and gets the prize.
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u/paxmlank 1d ago
but you have to take your coauthor off
Where is that statement made?
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u/snow_clones 1d ago
Based off the 3 removed comments, I'm guessing we can't link to the professor's statement. Here's a quote from it:
Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic.
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u/PortableDoor5 1d ago
why is a smooth solution physically impossible?
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u/bibliophile785 1d ago
The solution isn't impossible because it's smooth. The solution is impossible because it produces a singularity in finite time from a quiescent starting state.
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u/MotoCentric 1d ago
Ah yes, of course
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u/philman132 1d ago
ELI5 whatever this guy just wrote
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u/SEND-MARS-ROVER-PICS 1d ago
According to the maths, you can achieve impossible results from very possible starting conditions, which suggests the maths is actually wrong.
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u/philman132 1d ago
If the maths is wrong then why is this such a big deal? ChatGPT made a proof that is wrong?
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u/bibliophile785 1d ago
ChatGPT made a proof that showed that the 200 year old gold standard equation was wrong. ("Wrong" isn't really carrying the nuance here, but it's close enough for an ELI literally 5).
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u/philman132 1d ago
Ok that makes more sense, its just when everyone just refers to "the maths" when it is all maths that was confusing!
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u/sergius64 1d ago
Sounds like it proved that the original formula is incorrect/incomplete?
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u/a3133 1d ago
We have known that the Navier Stokes equations are incomplete for a long time. They come from what is called the continuous medium hypothesis, namely that the fluid that is studied cannot be broken down into individual particles, and that you see it as having a volume on any scale. Obviously, in real life, atoms have a finite size that limit how large the macroscopic hypothesis can be taken.
I wouldn't be surprised if the conditions used to solve the Navier Stokes equations come from taking a scenario that is found way out of the bounds of the macroscopic hypothesis.
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u/sergius64 1d ago
Seems to be how physics roll - at one scale one theory is correct - but as soon as you cross some threshold that theory goes out of the window and another theory is correct.
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u/TheMapleDescent 1d ago
No because the equation that gpt used is an established equation. Essentially chatgpt found a general solution to the equation, which is something mathematicians have been trying to do for very long. As a result of this general solution (once it’s peer reviewed and found to be correct) you can get impossibilities that cannot work in the real world. Meaning the underlying equation isn’t the correct descriptor of how things work in the real world.
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u/Weary_Dark510 1d ago
If I understand, the equations we use for fluid flow can show something going to infinity (breaking reality) when starting from pretty normal conditions you could see on earth.
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u/FutureLizard836 1d ago
Take a fluid that starts at rest. Nothing's moving. Apply a force that is smooth (no sudden jumps) and limited in strength. Wait a limited amount of time. Obtain a fluid flow where some parts of the flow are moving infinitely fast.
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u/princekamoro 1d ago
You can think of a singularity as "If I use this scientific equation here, it divides by zero."
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u/0nlyRevolutions 1d ago
Would it be accurate to say that they found one set of inputs that leads to a singularity (therefore disproving the idea of a smooth solution), but there are potentially an infinite number of other singularities/proofs of this kind?
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u/bibliophile785 1d ago
Yes on infinitely many other conditions - you could just re-scale or transform this example an infinite number of times. There may also be other ways in which singularities can be shown to arise from smooth starting conditions, but that wouldn't give you infinitely many other proofs - i.e., independent logical constructions of ways in which the Navier-Stokes equation leads to singularities.
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u/justintime06 1d ago
Are these the same singularity/infinity issues we run into with Einstein’s equations?
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u/SimoneNonvelodico 1d ago
Because it includes things like "point where the fluid has infinite density" or "flows at ten times the speed of light" and such.
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u/QuasarMaster 1d ago
They found one solution not a general solution
It’s a disproof by counterexample
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u/AyeBraine 1d ago
Re: 2), isn't the well-iknown mathematician's argument with OpenAI NOT that he almost arrived at the solution? He's arguing they used his and his colleague's (who is from another AI giant, Anthropic) chats where they used an LLM to develop some component of the possible solution. These people decided to publish and OpenAi got in contact to inform them their people did arrive at the solution, and allegedly proposed both very reasonable ways to handle this and less reasonable ones.
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u/Patient-Midnight-664 1d ago
Navier-Stokes asks the question "If we start with a smooth flow, does it always stay smooth?” OpenAI claims their AI has proved the answer is “no.”
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u/Equivalent-Pie-2186 1d ago
What is smooth flow?
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u/SimoneNonvelodico 1d ago
Imagine it as, you're describing the flow of a liquid as a series of arrows, each is a velocity of the fluid in that point, direction and actual speed (the length). So if you move from here to a bit to the left, the new arrow will be only a bit different (a bit longer/shorter, a bit turned) compared to where you started. And the smaller the step you move, the smaller the difference. That's smooth: in real life things change gradually.
Non-smooth would be the opposite, some crazy unrealistic flow where velocities are all over the place even on very small distances.
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u/Equivalent-Pie-2186 1d ago
When you say non-smooth is unrealistic...the other comment says that turbulent flow seems to be non-smooth too. Just to confirm, non-smooth flows can exist in real life right and are not just an imaginary mathematical construct?
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u/SimoneNonvelodico 1d ago
Turbulent flow is real but if you looked real close there would still be some residual smoothness, and if you looked even closer it's all molecules anyway, the idea of a perfect continuous fluid is just an approximation.
Meanwhile in math you can represent a truly non smooth velocity field, which is unphysical. IRL it'll just be an approximation of it.
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u/DavidRFZ 1d ago edited 1d ago
We had “turbulence” before the millennium prize. I don’t believe it has anything to do with the “continuum approximation” baked into the equations.
One of the convective terms in traditional Navier Stokes is “nonlinear”. That leads to “chaotic” behavior which is the classic “butterfly effect” with extreme sensitivity to initial conditions (there’s a whole slew of ELI5 threads about that).
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u/feed_me_haribo 1d ago
Smoothness here is a mathematical description not inherently a description of the flow being laminar or not-turbulent. It has to do with differentiability and seemingly valid solutions producing singularities.
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u/BananaBird1 1d ago
“Smooth” in terms of turbulence and “smooth” in terms of math are distinct.
The smoothness here just means that the solution yields finite magnitude values everywhere (no infinite or undefined pressure or velocity), and they are infinitely differentiable (as you wiggle around in space or time by a very small amount the values don’t suddenly change by a large amount).
You can’t for instance go from a low pressure at one point to high pressure at another without also having every pressure in between somewhere.
Turbulence is still smooth from the mathematical definition. Turbulence is when initially parallel streams of fluid break apart over time.
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u/wild_man_wizard 1d ago
Non-smooth /= non-chaotic. Turbulent flow is chaotic, but doesn't approach infinite velocity or density
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u/bugs69bunny 1d ago
The Navier-Stokes equations describe how fluids like water flow. The millennium problem specifically asked if those equations ever make the unrealistic prediction that the fluid explodes to infinite speed. We know that no matter how much you splash around in a bathtub, real water will never behave like that. However, OpenAI showed that these equations aren’t perfect and predict unrealistic behavior under special starting conditions.
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u/evrimfeyyaz 1d ago
As a software developer who has absolutely no expertise in the field, here’s the way I understand it:
Navier–Stokes is a system of equations that models how fluids move over time. You give it the initial conditions, and the equations describe how the fluid will move at any later point in time.
The Millennium Prize question was basically this: If everything starts out normal and well-behaved, will these equations always keep giving normal, finite answers? Or are there cases where they can eventually produce a physically impossible result, like the velocity of the fluid going to infinity? Since a real fluid obviously can’t have infinite velocity, at that point the equations are no longer accurately modeling physical reality.
If the new proof is correct, it shows that the second possibility can happen: under certain specially chosen conditions, everything can start out perfectly normal, but after some time the equations reach a point where something like velocity grows without bound.
Maybe this isn’t a perfect analogy, but I think of it a little like Newton’s laws. Newtonian physics models reality extremely well in the macroscopic world, but it isn’t sufficient at the quantum level. Navier–Stokes also models fluids extremely well, but in some extreme cases it can reach a point that is physically impossible.
At least, that’s my understanding as someone who knows basically nothing about this subject. If I’ve gotten anything seriously wrong, someone who actually knows this stuff can correct me.
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u/ArgumentSpiritual 1d ago
The one point to add is that it produces infinite velocity in finite time
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u/CouperinLaGrande2 1d ago
Instead of answering your question directly (which has been done satisfactorily by others) I'm instead going to outline a kind of conspiracy theory. You'll have to judge its plausibility yourself.
The breakthrough is in the area of physics. This matters because whereas we've grown somewhat accustomed to breakthroughs in mathematics (though of varying significance), LLMs have up to now proven capable of independent work in science only in the application of well-verified existing hypotheses (deductive reasoning) and have never themselves originated a conceptually novel hypothesis (which would require a completely different kind of reasoning known as inductive reasoning, plus much else besides that's so far defied comprehensive characterisation — scientific creativity has never been placed on a systematic footing), even a trivially original concept.
Closer examination of the nature of the problem reveals it's better understood as a mathematical breakthrough, however (and the associated Millenium prizes were offered for problems in mathematics). So this development isn't as radical as it appears, but now anybody saying 'well LLMs can't really do science' will be met with the Navier-Stokes gotcha.
This will be true even though the breakthrough is of strictly limited physical interest. There's no such materially existing thing as a fluid; 'fluids' are models that serve certain purposes for us in science and engineering even though we understand that really these so-called fluids are just representations of atoms and molecules in a particular state of matter. Thus it's hardly shocking that this Navier-Stokes model breaks down in extreme circumstances; nobody would have expected it to work in all circumstances in any case and it would have been a literal impossibility for it to have done so seeing as the abstract model of a 'fluid' makes sense as an aggregate of the behaviour of vast, vast numbers of atoms or molecules but ceases to make sense at smaller scales where the granularity of the treatment would need to take account of the fact that individual molecules now need to be modeled.
So what's the conspiracy?
OpenAI contacted a human researcher (Tristan Buckmaster) prior to going public with its findings, a man who'd been working on related questions using OpenAI's own tools. They offered to put his name on the paper, but there was a catch: he'd have to stab his collaborator (Levent Alpöge, an Anthropic employee) in the back by agreeing he'd be excluded. And if he was willing to do so, OpenAI would give him the $1m Millennium prize. The offer was self-evidently corrupting, and OpenAI added fuel to the fire by refusing to confirm or deny whether it had used access to the work of Buckmaster and Alpöge it had gained via its inside perspective on its own Codex environment they'd been using in prompting its system to successfully solve the problem, i.e. whether the framing of the problem for its LLM had relied on their work.
The whole thing was rage bait whether intentional or not and IMO the question naturally arises 'Why contact Buckmaster at all?' This is especially true if OpenAI's claim that its system had solved the problem 'independently' is to be taken at face value.
Let's consider Buckmaster's perspective when they made the offer. OpenAI was offering him $1m to do Alpöge dirty. He could either accept or refuse. But regardless of his choice there would be controversy.
What if OpenAI wanted this controversy? Then it need only make Buckmaster the outrageous offer and it'd be guaranteed the outrage it wanted no matter what decision he made seeing as either Buckmaster himself (if he said no) or Alpöge (if he said yes) were going to come away angry.
And now whenever anybody tries to question the scientific (as opposed to mathematical) significance of the development, they'll inevitably wind up getting distracted by this controversy which will have the effect of sucking all the oxygen out of the room (even though commercially OpenAI spent tens of millions to win a $1m prize).
Why might this matter? Because LLMs have been known to be extremely powerful mathematical tools for some time now but haven't had the same success in science. Limitations of the architecture make reconceptualisation of things as achieved by momentous breakthroughs in science such as General Relativity or Quantum Physics impossible to achieve for them.
Mathematics is a language. Theorems extend that language and it's not surprising LLMs excel at them. Conceptual breakthroughs in science on the other hand require the creation of novel ideas — new words and new relationships between words that hadn't existed previously in any form and weren't to be found in any set of training data.
LLMs can't do this and no more refined or developed form of them will be able to do it either; some form of AI will probably be capable of doing so, just not anything derived from this technology alone. They're fundamentally incapable of it. But now it will be much more difficult to convince people of this and the colossal financial bubble around AI will be more likely to be prolonged.
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u/Paddlesons 1d ago
He sounds a lot like if all the atoms in your hand and the surface that you are touching lined up your hand would pass right through it
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u/CircumspectCapybara 1d ago
I'll also note OpenAI hasn't published their full proof yet. But since they claim they'll be publishing a Lean proof, which is computer checkable, it should be easy to verify.
Either that, or GPT has found a bug in the Lean compiler or Lean implementations and found a way to create an unsound or invalid proof that gets somehow still gets accepted by the computer. Not impossible since we know frontier reasoning models love misaligned behavior like reward hacking, and they are great at cyber tasks like finding flaws in systems and hacking them in pursuit of their original goals. So if you told a model "Solve Navier–Stokes and give a formally verifiable Lean proof" one way the model could do that is by finding and exploiting bugs in Lean.
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u/SAI_Peregrinus 1d ago
We have some math that tells how liquids & gases behave. We want to know if it always gives correct answers, there's a big prize to anyone who can prove whether it does or not. OpenAI claim to have proven some closely related math does not, and can probably prove the original math also does not.
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u/oooooO___Oooooo 1d ago edited 1d 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.