Any problem that doesn't need to be solved exactly, but where one of many solutions that are 'good enough' are acceptable, or, more generally put, anything humans have been doing manually, is fair game for neural networks or even evolutionary algorithms (genetic included, of course). Neither humans nor the best autorouters 'solve' a board, but humans at least always manage to produce a solution that is 'good enough', and in certain situations autorouters can as well.
Neither we nor computers can ever find the perfect way to route a board, and for any given board, there IS a perfect layout, and perfect way to route it (for all the different ways you would define 'perfect', which could vary board to board of course). However, to find that perfect layout (and beyond that, know that it actually is the perfect layout) is whats called an NP Hard problem. NP Hard problems are problems where computers are just totally fucked. Any problem called NP Hard is basically a problem computers will never be able to solve algorithmically, regardless of how many more orders of magnitude CPU power continues to increase.
However, as CPU power increases, neural network/neural turing machine based AIs will quickly be able to come up with the same, fuzzy, "non-perfect but still pretty good" solutions humans can and have been doing for ages. This is by tackling the problem just like our brains do, and rather than finding 'the answer', it finds something that is not the answer, but is still good enough to be practical.
Cool post, ai_maker. I have experience in genetic algorithms to train neural networks, but in the context of game AIs (I made little worms that run/wriggle around and eat food and avoid poison - hardly anything that will revolutionize an industry any time soon hehe). The take away here is not the immediate practicality so much as that ai_maker is taking some first steps in thoroughly documenting proof-of-concept like posts in the context of electrical engineering and optimization, and in a few years time, I think stuff like that may become much more relevant to your standard board jockey, not just intel VSLI wizards.