So, none of the data you have to train with has a target value of 0? How would you expect any modeling tool to recognize when the alternative applies?
If you have severely incomplete data, nothing will improve it, except for getting more useful data.
Think of it like this: a neural net (like any other modeling tool) needs to learn from data. If all the data you give it only EVER has the value 1, then it will learn that the prediction is ALWAYS 1. ALWAYS, for any set of inputs.