Leonhard Euler

So the drama has been that these two mathematicians, Tristan Buckmaster and Levent Alpoge, which again, pretty sure I butchered his name, who works for Anthropic, were working on these problems already based on the work of some other mathematicians who had sort of pointed them in a direction, and they were using ChatGPT to do their work.

Um, they had found a solution for the Euler equation, um, which is basically a version, uh, which is essentially the same formula but with no friction, so no viscosity.

Um, and they were cleaning up the results of that work for publication.

Um, and then obviously they would move on to sort of think about the larger Navier-Stokes problem.

So what exactly happened? So he writes a PDF, explains, "Hey, here's the whole story about everything that happened." Basically, they go back and forth with OpenAI.

OpenAI says, "Hey, we would, you know, we would love to like share the results.

You know, you're gonna publish your results for Euler.

Maybe we co-publish the results for this other math problem.

I don't know what these words mean.

Supposedly, it is a blow-up for the Euler equations.

So basically, they're trying to figure out fluid dynamics.

[laughs] I know just enough about this to be dangerous.

End of A Garden of Old Delights by Lucy Maud Montgomery Read by David Wales

Influence of the Liberty of Spirits Upon Events by Leonard Euler 1707-1783 From Letters of Euler on Different Subjects in Physics and Philosophy Addressed to a German Princess Influence of the Liberty of Spirits upon Events If the world contained bodies only, and if the changes which take place in it were necessary consequences of the laws of motion, conformably to the powers with which they act upon each other, all events would be necessary, and would depend on the first arrangement which the Creator had established, of the bodies of the universe.

So that this arrangement, once established, it should be impossible for other events afterward to take place than those which happen in the actual order of things the world would undoubtedly be in this case a mere machine similar to a watch which once wound up afterwards produces all the motions by which we measure time imagine to yourself a musical clock such a clock once regulated all the motions which it performs and the airs which it plays are produced in virtue of its construction, without any fresh application of the hand of the master,

and

I, I clearly said that.

I was surprised to learn during the call with Tristan that they had only solved Euler and not Navier-Stokes." Um, Navier, I said it.

Navier-Stokes.

You did.

Mm.

"... because I felt it would be inappropriate for an Anthropic employee to author OpenAI's work.

Importantly, though, it was admitted that internal Anthropic models had been used in their proof of Euler blowup, so, uh, I therefore felt I couldn't consider Levant to be an independent academic.

Another option I tried to propose is to offer access to our internal models so they could try to finish the proof and bridge the gap between Euler and NS.

When likely we cannot rule out that de-identified data derived from their usage of products helped improve our models.

However, our proofs differ significantly, and even the precise results are different in the Euler case." And so a piece of nuance that I think people are really up in arms about is OpenAI, after saying they cannot confirm or deny, they basically offered to co...

like, co-author the work with just Tristan, not the OpenAI-- not the Anthropic researcher.

They suggested, "Cut him out.

But nobody's quite managed to get to a initial datum that blows up.

People have tried different things, and they've gone to Euler equations, and they've shown some evidence there, and we'll get to the story of [laughs] what happened here as we find out more of them.

They've tried to do things like physics-inspired neural networks, so DeepMind were really having a massive team looking at that, um, that moved a bit more analytically, and in fact, there was another release yesterday about that.

But this was considered to be a very hard, somewhat intractable problem, until it wasn't.

The viscosity term is the one that pushes back against it blowing up.

So as you get momentum there, the viscosity kind of pushes it back in.

So there's always a question of what type of structure do you need? So I believe DeepMind, for example, had a blowup on Euler equation where they had, like, this little wall, and this was the Chen-Hao one where they pushed and then it exploded because of the certain structure that they had.

And we've not had it for a more generalized one.