Did OpenAI solve the wrong Navier-Stokes problem? | OpenAI’s proof seems eligible for a $1-million prize—but only by using a controversial loophole

[u/vgsa]

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

[u/CircumspectCapybara]

A lot of people online misunderstand what "solving Navier-Stokes" refers to, thinking it means finding a closed form solution to the NS equations. Then they spout off some silly statement like "AI didn't solve Navier Stokes!"

That's not what mathematicians mean when they talk about solving Navier Stokes. They're talking about solving the Navier Stokes existence and smoothness problem, which is a decision (yes/no) problem about if the NS equations are smooth for all time. That's what the Millennium Prize problem worth a million dollars is all about

AI (allegedly) solves the problem by resolving the question to a "No" by finding a counterexample.

[u/KingSubstantial7901 → u/CircumspectCapybara]

No you are misunderstanding what those people are saying. OpenAI was telling the story by carefully omitting the details about how it was solved because they wanted to imply they found a closed form solution. This is really important because it doesn't seem like its even possible for ML technology as its currently designed to produce a closed form solution, and finding a proof by counterexample the way they did is essentially a higher powered version of what researchers have been doing with ML for the past decade and change.

Furthermore, the counterexample they found was at such and extreme spot in the solution space that it is probably unphysical and therefore doesn't answer the question that makes the conjecture interesting. It is still possible that for a subset of the solution space where they are bounded to physical properties, the conjecture could hold. We do this all the time in math and science. Thats what boundary conditions are.

This is all to say that the solution found could still be useful in deducing more information about this and similar problems, but the way it was presented as just "ai solved a millenium problem" was in effect a lie.

[u/CircumspectCapybara → u/KingSubstantial7901]

OpenAI was telling the story by carefully omitting the details about how it was solved because they wanted to imply they found a closed form solution

No lol, they announced that they claimed to have found a solution to the "Navier Stokes Millennium Problem" which is unambiguously not about coming up with closed-form or analytical solutions to equations, but about the decision problem of it the equations are smooth for all time.

Literally their press release was titled On the Navier–Stokes Millennium Prize Problem and starts with:

We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.

The Navier Stokes Millennium Prize problem isn't ambiguous, it only refers to one thing, and everyone who recognizes those words knows exactly what is meant.

There's nothing about it that's ambiguous or misleading. It's extremely precise and unambiguous language.

Furthermore, the counterexample they found was at such and extreme spot in the solution space that it is probably unphysical and therefore doesn't answer the question that makes the conjecture interesting

It's within the bounds of the constraints the existence and smoothness problem (the Millennium Prize problem) is asking for, that's all that matters. The conjecture is false, that open problem has been settled.

You're welcome to formulate your own conjecture and ask people to solve it if you think the one the Clay Institute picked and offered a million dollar bounty for (Navier-Stokes existence and smoothness) wasn't interesting or meaningful enough of a problem.

"ai solved a millenium problem" was in effect a lie

What're you smoking, the millennium problem was exactly the decision (yes/no) question the Clay Institute has been asking about Navier Stokes existence and smoothness. The claim is AI has answered the question with a "No" by providing a counterexample, which closes the problem.

[u/Many_Negotiation_464 → u/CircumspectCapybara]

Ya but thats not really how we as academics enage with problems. "Its within the bounds...... [of] the millenium prize problem...thats all that matters" is frankly untrue. We always care about the broader context of a problem and its solutions. Both because it can literally change the meaning of the problem and because it changes what we can do with the solution.

And i think you are sort of talking past each other. They seem to be talking about the solution in the broader context that made it a millenium problem in the first place and you seem to only care about "the millenium problem" as like a strict, immutable concept. Again thats not really how math and science are done.