Which algorithms have been proposed to learn the architecture of a deep neural network?
This predates the start of the "deep learning craze" in the mid-2000s, but for me the cascade correlation algorithm (Fahlman &amp; Lebiere, 1989; pdf) is the build-your-own-topology NN algorithm. I am not sure how popular the algorithm is in ML now-a-days, but it is still popular in cognitive science despite it's unrealistic grounding in biology. If you want intersections with genetic algorithms (like the one @vzn points out with more recent examples) then that has been done with CC as well (Potter, 1992; pdf).

You might also find the following questions on other SEs to be of interest:


Why do neural networks seem to perform better with restrictions placed on their topology?
Dynamically adjusting NN architecture: inventing the unnecessary? (also on MetaOptimize)