Thank you very much for the very nice introduction and I would also like to thank you very much

for the invitation. It's certainly a great pleasure for me to give a talk in this lecture

series. Now, I think we all know how tremendously successful deep neural networks and general

artificial intelligence is these days. I mean, we just need to look around us and see the impact

of it. I mean, for instance, self-driving cars, surveillance tasks, also legal issues. So in the

US, I mean, job applications are often pre-screened by these methodologies. And then the healthcare

sector, where, I mean, which unfortunately, I mean, these days is so tremendously important. And also

their deep neural networks and general AI has a significant impact, both on image and modalities,

but also on reaching decisions. But then, I mean, on the other hand, if we think about it from a

mathematics viewpoint, we quickly realize that there are very few theoretical results which

explain their success. I mean, two years ago, I mean, Ali Rahimi at one of the plenary talks

even spoke of deep learning is more or less alchemy, or can be considered alchemy these days.

And then maybe even worse, if we think of very sensitive applications, we see that sometimes

there's actually dramatic failure under very small perturbations. On the other hand, if you now look

at the model-based world where we are very familiar with and where we have worked for, let's say,

almost centuries, and we see that, I mean, these methods in very complex applications reach their

limits. So think, for instance, of the application of computer tomography, which will be a running

example for this talk. So what do you do there? Well, you have a human body and you compute nine

integrals through it. This gives you one slice of the sinogram and then you rotate the measuring

device. And so this way you compute the complete sinogram and from this you aim to recover

the interior of the body. But then, I mean, if you go to applications like what's called limited angle

computer tomography, where you cannot do the full rotation, what you see is you have only observations

in a certain area and you are missing a whole chunk of data. And so if you then reconstruct,

you observe that reconstructions look usually very blurry in certain instances like you see here.

And so the main problem with this is that somehow the data is too complex for very precise mathematical

modeling, because if you would like to now incorporate some prior information for your

reconstruction, you need to have a very precise model. And so in that sense, I mean, it seems

very sensible to applications like this and in general in imaging sciences to incorporate

learning methods and particular deep learning methods. And the path a lot of people take these

days is to combine the data-driven and model-based approaches. So data-driven, typically now machine

learning, deep learning, artificial intelligence. So in our case, this could be deep neural networks

based approaches. And on the other hand, let's say the more classical traditional approaches,

model-based where we have a lot of knowledge about and where we have also the means in a

systematic way to cooperate physical knowledge. And then a significant, the important question these

days is how to optimally balance both approaches. So shall we just throw everything over board,

the model-based approach and just use data-driven approaches or should we combine it in the same

more sensible way? And there are various opinions on that. I mean, some people say in maybe a few

years, the AI algorithms will be so good that they can learn the physics automatically. We then just

need a data-driven approach. Some people say we have the physical knowledge, so this is important.

Some people say anyhow, they need to be a balancing between both approaches. And so this is also the

message I would like to pass with this talk. And I would like to advocate to combine both

approaches in a sensible way. And we will see several examples of that.

Now, let me start with the classical side, because we will need this as a free knowledge

for then combining it with deep learning type of approaches. So I will now first focus a bit on what

has been done, let's say before the second wave of artificial intelligence and neural networks.

And then we will see how we can combine it with these new novel methodologies.

Now, let's talk about inverse problems. I mean, many problems in imaging are of that type,

are an inverse problem. Image reconstruction is one particular instance. We saw computed

tomography as one example where you have a forward operator. In the case of computed tomography,

we'll see this in more detail later on. This is the Radon transform. And so the task usually then is,