The Navier–Stokes Problem and the Agents

i want to come back from our listeners because they are not so deep in the whole topic but is maybe the domain mathematics the perfect domain for ai research ai driven research ai discoveries because it's it's a it's domain with clear structure and clear processes or do you know what i mean so that mathematic is maybe the best domain to prove something there

Yeah, I mean, maybe together with coding, that's one of the domains that can profit a lot from these post-training strategies.

So maybe to give a bit of an overview, so for people that not only OpenAI has not only announced that they solved this Navier-Stokes problem, but also they've uploaded 722 additional manuscripts where they claim to have solved some other math problems that were open they had to there were some problems three of the manuscripts they had to with withdraw and so on so there will be some problems and i think also the readability and so on this is difficult and some of them have this formalized lean proof some have not And this has led to a big discussion, I think, in the mathematical community and generally what's the impact of this and how will this field change? Why is it working so well? I think this is because you can always present kind of new problems.

so these language models learn first natural language and so on but also programming language and they can also learn of course on mathematical papers mathematical language is also a bit formal that we repeat stuff and introduce concepts and so on and now think a bit back to your high school you also got math small math from really simple ones to solve but basically your teacher could always come up with new ones you add this these numbers or in the text change slide things so this is very good for ai that you have basically infinite amount of training data you don't have you cannot memorize anything you always get a new problem and i think that's what they're doing with this for this model this that's unreleased that they presented a lot of mathematical problems and then it tries to solve it and then What the good thing is, you can verify if the solution is correct, right? If it's another field, for example, write an article about blah, blah, blah.

You cannot verify yes or no, if the article is correct or not.

But here, if the mathematical problem has been solved or whether the program in some whatever Python code or C++ code, you can check if the thing is correct.

also with the help of lean for example and that's a very good training signal for the neural network to learn from so these because they have these are two domains where you have verifiable rewards and that leads to very powerful models and then

Gonzalo, from your point of view, how much of the discovery came from the AI and how much depended on human mathematical intuition? Could a powerful language model have achieved the same results independently? What is your opinion on that?

So I think it is, so what is easy to think about here is the counterfactual where AI is not developed at the current level and mathematicians try to come up with this proof.

I think that would take decades for us.

I don't see any way mathematicians could be able to get there soon.

It's more difficult to judge whether AI without previous mathematical literature would be able to arrive to this proof because of course AI is trained on millions of papers.

So then which papers do we remove from the literature is a hard question.

Yeah, I think if it hadn't seen any or any of the recent results, I think it would not be possible.

And we see also that it can solve certain problems and some problems didn't progress that well.

So there are also fields or areas where it worked better and less so.

And also the energy invested was gigantic.

So I think at least 23 million people euros of compute were invested for this Navier-Stokes stuff.

And yeah, the price is 1 million euro, by the way, not euro, but dollars.

Yeah, but I think it will.

And also, of course, I'm very interested in Gonzalo's opinion.

I think it will have an impact on mathematics, how they will proceed.

I think Terry Tao said it will become more industrialized and larger groups working maybe together with AI.

We've seen other scientific fields like molecular biology where that is more industrialized.

There are big consortia working on problems.