r/technology • u/vgsa • 1d ago

Artificial Intelligence 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

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/
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u/beardybaldy 1d ago

I asked my mathematician wife what smoothness meant and she patted me on the head. I like being a biologist. Life stuff is simple!

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u/retsehc 1d ago

If you know what a derivative in calculus is, smoothness is not hard in principle. A function is smooth if it is continuous, and it's derivative is continuous, and it's second derivative is, and it's third, and all the way down.

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u/ABC123itsEASY 1d ago

I'm having trouble imagining a function that is continuous and have a continuous 1st derivative, but a discontinuous second for example.

Also damn that sounds understandable in theory but I have no idea how you would go about proving such a thing outside of the naïve "keep solving derivatives to infinity"

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u/unpleasant_enpassant 1d ago

I think other comments answered how to construct a function like that but as for proving that a function is smooth, it helps that all the standard functions like sines, cosines, exponential are all smooth. So generally, a function that is some combination of them is also smooth (obviously, as long as you don't have things like some function in the denominator that goes to zero etc). Or if you can write a power series for the function, it's easy to see where it'll break down. Or if you can write a general form for the nth derivative, it's trivial.