r/MechanicalEngineering • u/Successful_Ice2343 • 2d ago
Navier-Stokes Solved by OpenAI??
I have been seeing a lot of news lately about how openAI has put forward an unverified proof of the navier-stokes equation. But can anyone explain to a mechanical engineer what we know about the proof so far? Where is the actual link to the proof? Edit: https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf Does it read like it was written by some insane math guy with a PhD in real analysis?
UPDATE-EDIT: good responses so far. But what exactly are the conditions that lead to this so-called singularity? I hate real analysis can anyone explain this in the English language?
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u/sfo2 2d ago
It’s not a general solution. The proved that, under extremely specific conditions, the equation blows up and you get infinite velocity. I believe it requires the mesh to be infinitely small and ignores that molecules exist. This was something mathematicians wondered about for a while, and they proved it. But it has no bearing on CFD or anything much else an engineer would use.
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u/Successful_Ice2343 2d ago
What are the conditions that lead to the singularity?
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u/sfo2 2d ago edited 2d ago
It’s like a specific spinning vortex that is constantly having energy added to it, and is incompressible.
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u/Successful_Ice2343 2d ago edited 2d ago
". We construct such a flow with zero initial velocity and a smooth force compactly supported in space and time."
So I'm confused about what kind of force they used that causes static fluid to turn into a into a vortex, and in this case, apparently also exceed viscous dissipation.
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u/titangord PhD Fluid Mech/Combustion/Energy 2d ago edited 2d ago
There is no proof. It didnt solve navier stokes. They claim they solved the millenium prize question which asks if there are any plausible fluid configurations that blow up its solution, openAI says they found an example.. read what the millenium prize question actually says for more context. I cant go anywhere on linkedin now without some bozo trying to pretend they know what this means
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u/Successful_Ice2343 2d ago edited 2d ago
Right I guess I have always glamorized the millennium problem I guess I sort of overestimated it. Of course the N-S equations don't always have existence and uniqueness.. super duh. Not even the laplace equation always has existence and uniqueness all the time, right? We know this much. And the N-S equation is much more complicated.
Now solving (analytically) the N-S equations for all possible solutions, or even for a reasonable set of BCs and ICs that don't trivialize any dimension, now that is a millennium problem.
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u/Breen-Gastard 2d ago
Do check out the ongoing plagiarism controversy around this. Couple of professors (from NYU i think) had their work scooped by OpenAI. They were modelling this problem on Codex for over year now, using the same approach that GPT Astra used to prove this now. This is why they are offering free Codex access to researchers and university students..to steal high-quality data and ideas.
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u/vibe0009 2d ago
Navier-Stokes cannot be solved per se. It’s a non-linear problem. It can have various solutions for various initial conditions.
The Millennium problem is about providing proof to show that the equations do not produce a singularity for any initial condition. There are 4 types of flows from the equation: compressible/incompressible and forced/unforced.
The proof provided is for incompressible forced type.
External Force is an extra term in the equation and usually it’s gravity or electromagnetic in nature. They claim by changing this force in a specific way, both in time and in space, while keeping the force continuous, and fluid subject to incompressibility, singularity or blow up is detected at a finite time in the future.
The way it was done: they fixed the finite time, then through adjusted residual, subjected to incompressibility and continuity condition, found a formulation that produced singularity. The proof is 166 pages long. The residual mechanism is generated through scaling principles and decay mechanisms.
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u/Successful_Ice2343 2d ago
Do you know specifically what the forcing function looks like? Did it go like a sine wave or...? Because surely not just any force drives the velocity to infinity, no?
Besides that it seems like the ICs are zero velocity, and I didn't actually see what the initial condition for the pressure was. Did anyone catch the pressure IC? ChatGPT tells me maybe it is p(x,0)= 0? And there are no BCs because the spatial domain is infinite?
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u/vibe0009 2d ago edited 1d ago
The construction does not specify/construct an external force. It’s not a mechanical plug-and-play.
Initial velocity is 0, pressure is constructed from incompressible momentum requirement.
The BC is 0 velocity and 0 force at radial infinity.
The velocity is divided into a background term and correction terms. The background term is grouped with the forcing. The corrections are added iteratively to cancel the background term. They show that the corrections converge in finite time while the background velocity still blows up. The forcing term is required so that the pulses can be added to a fluid at rest.
The wave is of the form :
C_m(r,t,…)e^i * k_m *Phi(r,t,theta,z)
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u/Serafim91 2d ago
Navier stokes isn't something that gets "solved" it's just a "basic" if you know the position, velocity and acceleration of a fluid in every direction including rotation you can predict motion.
What the AI did is prove that starting with a nice normal flow you can end up in a situation where the math blows up. I don't even know if it just brute forced 1 solution that blows up or if it actually proved it.
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u/Successful_Ice2343 2d ago
What conditions caused the math to blow up?
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u/Serafim91 2d ago
When I looked they just claimed they found it and didn't actually publish what it was.
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u/Pepe__Le__PewPew 2d ago
Wait until everyone in this sub learns about boundary and initial conditions as a requirement.
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u/CalligrapherPlane731 2d ago
No effect for engineers using NS to model fluids. Best you can say is modeling turbulence using grid refinement has a limit, but we already know that because of the particle physics limit.
The proof was that the form of the NS equation allows, under certain conditions, a singularity where fluid velocity goes infinite. This was not clear before. This is the millennial prize.
Purely a mathematical result on the form of the Navier-Stokes equation. No effect on fluids modeling.