Can a deep neural network be used for mining bitcoins? Why or why not? (The reason I ask this question is because I know GPUs can be used for mining.)
That's a very tenuous connection!
A deep neural network is basically an element from a group of functions that are good at approximating another function whose value is given only on a subset of possible inputs (i.e. by some data and not by some rule/formula). The problem of finding this function can be solved by algorithms, such as gradient descent and its versions.
Mining bitcoin, as far as I know, is about exactly inverting some hash function by trial and error. The point is that we have not got a better algorithm than trial and error and that approximations are not sufficient, it has to be exact, and that makes it costly, which makes the system secure. So no, there is not much similarity between the two.
That in your mind the two are connected by GPUs just means that:
For bitcoin, the calculations required for mining are very simple, requiring little memory, but they are numerous, and GPUs are quite good at running many different instances in parallel, so they can try many at one time. The best hardware for mining are actually for mining tailor-made application-specific integrated circuits (ASIC). Those won't be able to/good at train/ing a deep neural network.
For deep neural networks, I'm not even sure that GPUs are generally useful or required for training DNN. They are very useful for convolutional neural networks which are used in image recognition and manipulation (and also for video) but that is because images with their many pixels require highly repetitive computation which can be nicely parallelised on a GPU because that's what GPUs are made for.