Alright.

Thanks for the introduction.

My key, the content I was planning to talk about are these differentiable simulations.

In the meantime, since I agreed to do the talk, I slightly at least shifted the goal

and I want to also put quite some emphasis, probably it's always half half, I think, but

not on the fusion modeling.

So I think especially the combinations, actually a very interesting topic that keeps my group

quite busy here at the moment.

And I think it's an exciting direction within this larger deep learning field.

Hence, this coupling between the two.

Alright, let me start very broadly.

So I used to start these talks by trying to convince people that deep learning is worth

looking at at all.

I'd be happy to go into more detail here, but by now I'm seeing across many fields that

there are some consensus that it's a tool that's worth looking at, that's not completely

off the hooks.

And in a way, right, it's just another tool in a toolbox and visually you can't imagine

it.

It's a shiny new hammer that's more or less pretty quickly dropped out of nowhere seemingly

and offers a lot of questions and interesting research directions figuring out how to

exact the employer and where it pays off most.

But at least I think by now across many people we have a consensus that it's at least

worth looking at as one alternative of numerical methods.

And at the same time, working with simulations, I'll be focusing on fluids a bit as already

visible here on the right.

Many of the physical systems we study actually have uncertainties.

And here's this fluid smoke example in motion closely related, different interfaces and typically

model as two fluids are liquids.

Classical applications could be airfoil, flows, lift and drag calculations.

Many settings here are fairly well understood, but have uncertainties, be it in terms of

the inherent randomness of the process or the unpotentially incomplete or ambiguous measurement

that we have in certain states, typically dubbed as as aliatory uncertainty.

And this goes in contrast to the uncertainties and errors that we have in our models and

representations that we form in the computer.

That's typically the epistemic uncertainty.

And in the following, I actually want to focus on the first kind on these aliatory uncertainties.

So the ambiguous states and the uncertainties that we have in the physical descriptions and

not so much on the second kind.

And that's in a way a bit closer to the actual learning with more data and refinements

you can reduce the uncertainties there, but I'll be assuming that's basically, for example,

we have an incomplete observation of a flow and then we are not sure how exactly this measurement

will end up in the future a few seconds down the road or how exactly this on average

would lead to drag that we experience in an airfoil, so cases like this.

All right, in this context of uncertainties and randomness, I directly want to introduce

the basics of these diffusion models.

So let me start with the basic equations here.

We have some quantity of interest, typically Y and function f of x that that constitutes

these.

We typically assume in the following we approximate this with a neural network with some

way it's data and we have some approximation error.