Evolving artificial neural networks for solving NP problems
No. This direction is unlikely to be useful, for two reasons:


Most computer scientists believe that P $\ne$ NP.  Assuming P $\ne$ NP, this means there does not exist any polynomial-time algorithm to solve any NP-complete problem.  If you want your neural network to solve the problem in a reasonable amount of time, then it can't be too large, and thus the neural network will itself be a polynomial-time algorithm.  It follows that if P $\ne$ NP, neural networks cannot efficiently solve any NP-complete problem.
Neural networks aren't "magic".  They are a way of trying to find patterns.  For some problems where there are strong enough patterns to be found, and the patterns can be learned from a reasonable number of examples, they might be effective.  But they're not magic fairy dust.  Just because you can set up a neural network doesn't mean that backpropagation will necessarily find a good way to solve your problem.  It might be that there are no patterns to be found, that the patterns can only be discovered with an unfeasible number of examples, or that patterns exist but the neural network training procedure isn't able to find them.


Neural networks are just another form of machine learning.  We could make the same remarks about SVMs or random forests or linear regression any other form of machine learning.  Neural networks aren't some kind of magical silver bullet that solve all machine learning problems.  They're about as effective as other machine learning methods, or for some kinds of problems, maybe a little bit more effective, but they're not magic.

Sometimes I run across people who have heard only a little bit about neural networks, and they walk away thinking that neural networks are the answer to everything -- maybe because they heard that "your brain uses neural networks too", or they saw some very cool application (voice recognition or something).  But don't be fooled.  Don't believe the hype.  Neural networks are a useful technique, but they're not going to enable computers to solve NP-complete problems, or to beat the Turing test, take away all our jobs and replace humans with computers.  Not anytime soon, anyway.  That's just science fiction.