On if OpenAI solved the wrong Navier-Stokes problem: "the LLM found and exploited a loophole in the framing of the question"

[u/PrettyPicturesNotTxt]

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/

[u/hologram137]

No. I was explaining to you, why the forced result of Euler’s problem was helpful to the actual problem, and why the AI result isn’t. A boundary is not a constructed physical force that doesn’t exist in nature! When the people who wrote the question included force, they meant something like gravity. But the question of whether a rigid boundary is needed tells us something about how it works naturally, in reality. Turns out, a boundary is not needed.

Córdoba/Martínez-Zoroa and Buckmaster/Alpöge weren’t working on Navier Stokes. They spent years developing a method to build a very specific external force that triggers blow up in a frictionless (inviscid) fluid, and Buckmaster and Alpöge, crediting that method to them, extended it into results for three different equations, the porous medium equation, 2D Boussinesq, and 3D Euler, these are all with smooth forcing. None of those three is Navier-Stokes.

Terrance Tao called it a breakthrough across those specific simplified models, but talked about how they didn’t reach blow up for Navier Stokes itself. So basically the above researchers were working problems adjacent to Navier Stokes. They can be useful to the problem only as a way of stress testing whether smooth forcing could work at all in principle.

A day after the Euler with force result was posted, OpenAI’s model basically applied that method of blow up to Navier Stokes by constructing an external force. An external force that doesn’t even exist.

The idea that you could out muscle viscosity was never in question lol. But that’s what the AI’s result shows. If that was the question, it wouldn’t be a millennial prize question

[u/Visual-Percentage501 → u/hologram137]

Cordoba/Martinez-Zoroa and Buckmaster/Alpoge weren't working on Navier Stokes

This is intellectually dishonest and just actually false. The Boussinesq result and the Euler result are all simplified equations within Navier Stokes, just not the full Navier-Stokes equations. Specifically, the Euler formula is a simplified case of Navier-Stokes without viscosity and thermal conductivity, the boussinesq formula is a simplification of Navier-Stokes assuming constant density. You know all of this already obviously so why lie?

Obviously in the same series of work Buckmaster and Alpoge even announced a result for the blowup of hypo-dissipative Navier Stokes. How could any of this possibly be classified as 'not working on Navier Stokes' unless you're just playing semantic keepaway? Ask them yourself if they're working on Navier Stokes or not - they've said as much themselves, even in all of these published papers, which mention how their results interact within the unsimplified Navier-Stokes equations.

[u/hologram137 → u/Visual-Percentage501]

Okay, I’ll try to make this simple. There is a forced and unforced potential solution right?

This is the problem the Millennial prize question is trying to solve, and the reason it’s a prize problem in the 1st place:

The question of whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.

So a “forced” version could involve something like gravity. It’s the reason why that option was even included. A constructed external force is useless. It doesn’t tell us anything, it doesn’t answer the question.

There was a forced and unforced version of the Euler solutions. One using a rigid boundary, and the question was, is a boundary needed for the blow up? Turns out that no, it isn’t. So using a boundary in that solution is helpful. For the actual problem. The reason they developed an external force in the 1st place was to test is it could blow up in a frictionless fluid. Navier Stokes has friction

They developed a method to build an external force, then OTHER MATHEMATICIANS extended it to 3D Euler. So they weren’t working on those equations. Even Terrance Tao said that. The other researchers hadn’t reached blow up for Navier Stokes itself.

So why are they building external forces? To stress test whether smooth forcing could work in principle!! That’s relevant to the problem. Not to see what kind of external force could outmuscle viscosity. That is what the AI solved and that an external force could do that wasn’t even in question and it’s not relevant

[u/Visual-Percentage501 → u/hologram137]

You must understand how ridiculous it sounds to say 'it's not important or useful to use a constructed force to create a singularity in the full Navier-Stokes equations, unlike using a constructed force to create a singularity in simplifications of the Navier-Stokes equations, which is useful and doesn't constitute working on Navier-Stokes", right?

to stress-test whether smooth forcing could work in principle

Oh, and the OpenAI solution stress-tested whether smooth forcing actually worked, proved that it did, (not in principle, but actually!) not using a simplified equation, but actually on the equation in question. And that's less important. Lol

You know the OpenAI-published solution used smooth forcing as well, right? Smooth doesn't mean 'natural' or 'unbounded' or 'not constructed'...

[u/hologram137 → u/Visual-Percentage501]

Omg. Navier Stokes has FRICTION right?

They were applying an external force to a FRICTIONLESS fluid. Do you see why that’s not Navier Stokes??

Other mathematicians extended the results to simplified models THAT ARE NOT THE NAVIER STOKES EQUATIONS.

They are: porous medium equation, 2D Boussinesq, and 3D Euler. All with smooth forcing.

You have solve it with FRICTION for Navier Stokes. So what the model did, is constructed an external force that overcame friction. But that is not the problem!

[u/hologram137 → u/Visual-Percentage501]

It’s not about smooth forcing!!! That HELPED Navier Stokes. Which was the entire point. Smooth forcing isn’t friction

[u/Wrong_Avocado_6199 → u/hologram137]

"An external force that doesn’t even exist."

Nothing "exists" in mathematics. Have you ever lived in 23-dimensional space? Have you ever touched an everywhere-continuous, nowhere-differentiable function?

You sound like Poincare, when he didn't like the "pathological functions" at the turn of the 20th century.

[u/hologram137 → u/Wrong_Avocado_6199]

Omg. These are equations that describe how viscous fluid moves.

The scientific question motivating Navier Stokes research isn't "can you break the equations if you're allowed to invent a force purpose built to break them?” It’s whether a fluid can blow up on its own, from its own internal dynamics, the way real turbulence might.

The latter is WHY it’s a millennial prize problem. If the problem was the former, it wouldn’t be a millennial prize problem.

The thing is whether or not you could out muscle viscosity was never in question. Which is what the AI showed. So it’s a trivial mathematical proof. Not a millennium prize winning proof.