**Abstract.**

Variational methods are a quite universal and flexible approach for

solving inverse problems, in particular in imaging sciences. Taking into

account their specific structure as sum of several different terms,

splitting algorithms provide a canonical tool for their efficient

solution. Their strength consists in the splitting of the original

problem into a sequence of smaller proximal problems which are easy and

fast to compute.

Operator splitting methods were first applied to linear, single-valued

operators for solving partial differential equations in the 60th of the

last century. More than 20 years later these methods were generalized in

the convex analysis community to the solution of inclusion problems and

again more than 20 years they became popular in image processing and

machine learning. Nowadays they are accomplished by so-called

Plug-and-Play techniques, where a proximal denoising step is substituted

by another denoiser. Popular denoisers were BM3D or MMSE methods which

are based on (nonlocal) image patches.

Meanwhile certain learned neural network do a better job. However,

convergence of the PnP splitting algorithms is still an issue.

Normalizing flows are special generative neural networks which are

invertible.

We demonstrate how they can be used as regularizers in inverse problems

for learning from few images by using e.g. image patches. Unfortunately,

normalizing flows suffer from a limited expressivity. This can be

improved by applying generalized normalizing flows consisting of a

forward and a backward Markov chain. Such Markov chains may in

particular contain Langevin layers.

We will also consider Wasserstein-2 spaces and Wasserstein gradient

flows, where the above Langevin flow appears as a special instance. We

will discuss recent developments for estimating Wasserstein gradient

flows by neural networks.