The "correctness" of neural networks is a vasy field of active research, with many prominent minds in deep learning as well as traditional CS theory working on it. We have many results for small scale networks, but I don't know of any results that can "prove" the correctness of an image classifier, for example. After some point, the correctness of such methods becomes very ill-defined, since unlike the normal programming world where everything is mostly in binary, here you will have to answer questions with some variation of "this is 98% likely" with no scope for 100% certainity.