So for instance, when dealing with the problems of classification of data and supervised learning,

we have shown how one can use residual neural networks in a given dimension D in order to,

you know, layer by layer, make information evolve and succeed on, in particular, classifying

in a successful manner so that eventually these can be later used for generalization, right,

to classify new data that are unknown. So as we have seen, basically, the way we, you know,

the perspective we adopted was that of nonlinear dynamical systems, either continuous in time

or discrete in time governed by sigmoid nonlinearity, right, and manipulated, regulated through

parameters, controls, depending on time that were taken into consideration the possible need of,

you know, shifting the interface of the neural network in the opinion space or to better classify

or also to choose the cutting hyperplanes that will determine what is the moon region and what is

the one that is frozen and then also re-entering the wind along which the moving region will move,

right. And this was done in dimension D, right, with a given, say, width for the network.

But you see in view of the idea, you see that this idea can be easily extrapolated also to the context

of neural networks. Of course, this is a very general idea. Details have to be worked out later,

right. So the analysis might even be complicated. So the analysis of the perfectly matching layers

that was actually introduced for the Maxwell's equations in Elastia, in the system where this

is actually a system of equations is not simple. The idea is quite generic. As we said, given any

system inside this box, any process, any continuous media evolving inside this green box,

we can always set up a little, say, layer around. And in this layer, the equation can be tuned

arbitrarily because after all, right, once we solve the equation in the big domain, we are,

you know, neglecting everything that has been added here artificially and focused

simply on the equation and the solution inside the original domain. So then even if originally,

you know, we have added this artificial term because this artificial term is only active on

the exterior layer, we can focus, right, and restrict the solutions to the original domain

so that we preserve the true equation. So as I said, then the analysis needs to be,

you know, justified. But, you know, computationally, this is something that is actually

extremely useful and not so hard to implement. And one can, of course, soon realize, you know,

impact that such a damping mechanism has on the behavior of solutions. So this idea could be also

implemented in the context of deep neural networks. So how you will do it? I mean, you have to,

if I will ask you, okay, can you extrapolate this here that originates on the wave equation

or wave-like equations by exterior layer of damping material? Could you generate it to the

context of, you know, neural networks in which we are, you know, moving from one, say, layer

to another, to another, right, to another, so to generate a map, right, that's out of,

that's our original problem, right, out of a cloud of very mixed data, right,

is able to really, you know, place

the data corresponding to each label well separated in the arrival space?

How would you extend this idea to that setting? Note that the phenomenon could be similar here

when we are advancing, we are marching on K, which is the pseudo time on our deep neural network or

the real time in case we decide to make it an ordinary differential equation to work with neural

differential equations, right? So, but the phenomena could be similar. So for instance, here,

you know, we know that the wave equation, right, is oscillating inside, but when getting to an

artificial wall, waves will bounce back, generating, you know, phenomena that are spurious.

This is when we decided to continue, right, the, you know, the wave equation outside,

the wave equation outside, let waves get into the other frame, but add the empty, the damping there

so that they never come back, right? Of course, this idea can also be implemented in the context of,

of neural network, right? You could always say, okay, no problem, I add one extra layer here

and one extra layer here, right? I make the network a bit wider. I design the coefficients

for the wide neural network, but then eventually I try to project into, you know, the original width,

right? And in this way, the fact that I am allowed to add an extra layer up and down, right, this will

allow me, for instance, to introduce many dissipative effects here that will have an impact also