A couple of disclaimers. So first of all, I apologize for changing the title and I did

that for two reasons. So first, because I was here already invited by Enrique last year

and anyway half for like 10 years or so I have been working on this digital twins and

reservoir computing and all these randomization methods for the learning of dynamic processes.

But in any case, I see that well, I realized that much of his group is already here. They

heard this already. And besides, I gave this talk for the first time a couple of weeks ago

in over Wolfack. It worked pretty well. So I said that, well, maybe it's a good idea to

give it here too. But having said that, you're a bit my guinea pigs, you see that there are too

many slides. I don't know how this story pan out, especially at this time of the day. But in any

case, so this is a learning talk. So let me go, I will go slowly, slowly, because I know that we

are a very heterogeneous audience here. And anyway, we'll go slowly, slowly. And there is

a more significant learning part that in previous talks, it is somewhat related to Professor Concas

talks. So ultimately, what we will be doing here is we will be solving an inverse problem. You

could see it like that. But instead of using the traditional analytical methods associated mostly

to PVE people, what I will be doing is a learning approach. So my tool of choice here will be RKHS,

so it will be Recurrent Kernel Hilbert Spaces. I will tell you a little bit what that is and why

that is important. But in any case, I will start by introducing what my context is, because not

everybody here is associated with or is familiar with Hamiltonian systems. So I will start by giving

you a brief idea of what we are doing here, why it is important. I will be explaining to you the

tools that I will be using. And then I will do a bit the learning parts. So I don't want to overwhelm

you with pack bounds and convergence rates and the usual statistical learning stuff. But in any case,

I will try to convince you that the problem is first important, that the technique that we are

using is pertinent, and that actually in numerical experiment this gives you something that can be

computed even in global situations, because we haven't seen much of it in this conference. But

you see, for people who are into mechanics and geometric mechanics in particular, the non-Euclidean

part of this whole story is very irrelevant. And that's something that when you do geometric mechanics,

we have a machinery that is ready to be used. But when it comes to using all these things for

learning, it's not so clear. So actually, this is like the third try that I, together with my

collaborators, which by the way I didn't mention, and they are behind the camera. So I have to say

that this is work with these two great, two young individuals. So in you, he's a postdoc in my group,

and Daiging is my student. They will probably be in the job market next year, so you may want to

remember their names. And in any case, so then let me get started with something very simple, right?

Just in case you didn't see this thing in high school, so the systems that we will be trying to

learn here are Hamiltonian systems, right? Hamiltonian systems are the systems of, if you

want, physics, right? So they are characterized by this Hamiltonian function. Then you have the q's

and the p's. The q's are the positions. The p's are the momentum, right? So this is the simplest

version of a Hamiltonian system that you can encounter. But this is not actually what you

encounter in many applications, because Hamiltonian systems, formulated like this,

you can use them for interacting particles in Euclidean space. But in reality, for other

physical systems like robotics or fluids or plasma or elasticity, you have something which

is more sophisticated. So you need to go higher in mathematical tools that you will need. So when

you go non-Euclidean, so think for example of a robotic arm that has two joints, right? So there,

your phases space are not going to be q's and p's. They are going to be, well, points in the

cotangent bundle of a two torus. You see that immediately even for very simple situations,

this non-Euclidean situation kicks in. And then how did you do mechanics there? Well,

you need to introduce something that we call a symplectic manifold. So it's a manifold endowed

with a closed non-degenerative two form. So you need to take some, to do some differential

geometry here. And then based on the using this non-degenerative of this form, out of the derivative

of the Hamiltonian or the external differential of the Hamiltonian, the Cartin sense, you end up

having a well-defined vector field, which is a section of the tangent bundle in this manifold