Remove neural nets from the picture and ask yourself what you're doing first.

You have the outputs of a non-linear function, and you want to corresponding inputs to this function. This is akin to finding the inverse function for the problem.

Now bringing in neural nets into the picture, I believe it is quite hard to analytically compute an inverse to it's underlying function. I'd believe it's just easier to train a new neural net representing the inverse. Of course, in certain domains, this might very well not work(image classification for example).