Can a neural network be used to find a functions minimum(a)?
Back-propagation works by minimizing the error.  However, you can really minimize whatever you want.  So, you could use back-prop-like update rules to find the Artificial Neural Network inputs that minimize the output.
This is a big question, sorry for the short answer.  I should also add that my suggested approach sounds pretty inefficient compared to more established methods and would only find a local minima.
I had been interested in neural networks for a bit and thought about using one in python for a light project that compares various minimization techniques in a time domain (which is fastest).
Then I realized I didn't even know if a NN is good for minimization. What do you think?