Can a neural network be used to find a functions minimum(a)?
They're pretty bad for the purpose; one of the big problems of neural networks is that they get stuck in local minima.  You might want to look into support vector machines instead.
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?