Is it possible to train neural network to draw picture in a certain style? Is it possible to train neural network to draw picture in certain style?
(So it takes an image and redraws it in a style it was trained for.)
Is there any approved technology for such kind of a thing?
I know about DeepArt algorithm. It is good to fill main image with certain pattern (for example, vangoghify image), but I am looking for something different - i.e., for example, making cartoon in a certain style from the input portrait.
 A: This is a pretty difficult problem to solve. You can see some examples here on how a cartoon style, e.g. from the Simpson's has been applied to an image. 
A cartoon image generally doesn't have the structure that gives this artsy effect. The easiest way to try to apply this in some way would be to have a face-tracker, and then try to align two faces, e.g. a cartoon face and a human face, and then apply this. That might get you somewhere, but it might also look weird. You might then annotate landmarks in the images to help further and do a non-rigid registration before this. This is still somewhat a shitmix solution, but the closest I can think of that could work for faces.
Edit:
The comment by @TannerSwett adds something to this, it is potential to go onto some artists webpages and try to find their illustrations and try to learn "their" style. I still do not think that will satisfactory or yield enough data, but that would be an interesting thing to test. There is no generally available solution right now, but I think that are definitely some people working on this, and we will see better results soon. 
I think that maybe the way to go is not the artistic neural network approach. Maybe it is better to have a network that can classify objects in an image and then learn the correspondences between the objects and their cartoon counterparts, then blend the results in some meaningful way.
A: It shouldn't be too complicated to do. 
Haven't read the article mentioned, here's my recipe:

Variational Auto Encoders
Online demo with morphing faces:
  http://vdumoulin.github.io/morphing_faces/online_demo.html
and https://jmetzen.github.io/2015-11-27/vae.html for teh codez.

Basically, this gives you a way to parametrize the 'style' in your case, for example let's say how wide or fuzzy should the brush stroke be. Stuff that depends on the particular style you are trying to emulate.
In the example above different 'morphed' or 'imagined' faces are a function of the parameters in the latent space. In the image below that would be what you get by changing stuff at the 'code' level.
Here's the basic idea: original image left, stylised version of the same image on the right:

Now, in theory, if you would train such a model on a normal image and a stylised image as a target and add convolutions, you should be able to learn the kernel filters that correspond to the type of "brush strokes" that the artist uses.
Of course, that means that you need to have a few examples of images in both original and stylized versions. Such a dataset would be nice to donate to the community - if you end up doing this I'd be very keen to see this sort of work.
Good luck!
The wiki article on auto encoders would be a good starting point:
https://en.wikipedia.org/wiki/Autoencoder
A: There is a relevant paper: LA Gatus, AS Ecker, M Bethge, 2015, A Neural Algorithm of Artistic Style. Quoting from the abstract,

Here we introduce an artificial system based on a Deep Neural Network that creates artistic images of high perceptual quality. The system uses neural representations to separate and recombine content and style of arbitrary images, providing a neural algorithm for the creation of artistic images. 

Here is  Figure 2 from this paper:

There is also a very popular open-source implementation based on torch here which is quite easy to use. See the link for more examples.
Keep in mind, that the computations are heavy and therefore the processing of single images is the scope of this work.
Edit: after checking your mentioned DeepArt project, it seems it is using the same techniques. I'm not sure why this is not what you want, because the concept of style-transfer is as general as it gets.
