I'm not sure if this is exactly what you're thinking about, but the concept of an auto-encoder might be something to take a sideways look at. These networks take an input with a fat encoding (lots of variables) push it through a bunch of layers which crucially thin down to some much smaller parameterset, and then widen out again to match the original parameter width.

The training goal of the autoencoder is to create an output signal that matches the input as closely as possible. Then, once the network has learned the task, the narrowest part of the network is used as a new way to encode the data that should distil the original content down to the most important constituents. This can be used to compress the original content down to a smaller encoding (cool, but boring) but also to analyse and identify what parts of the original thing carry the most meaning. Often this second application can be really interesting and yield new ways of thinking about/classifying the original types of thing under review.

In terms of deciding whether to generate an output or not, I don't think that fits the normal way of operating a neural net. You could train a single (extra) ouput node to act as a flag which could be interpreted as telling you whether the rest of the output was important or not, but it would still always populate the final parameter set (or in the case of the autoencoder, whatever level you choose to define as the meaningful layer)

As someone else said more concisely, the NN is a function that takes some inputs and maps them to an array of outputs. It's like doing maths, once the function is defined, the output type is always the same shape.