neural net with external nonlinear function on output

I am interested in trying NN in a perhaps unusual setting.

The input to the NN is a vector. The output is also a vector. However, the training data and error is not computed directly on this output vector, but is a (nonlinear) function of this output vector. So at each epoch, I need to activate the NN, find an output vector, apply this to my (external) nonlinear function to compute a new output vector. However, this new output vector is of length 1 and the error is computed based on just this single output.

Some questions:

1. Is this something that NN might usefully do?
2. Is this a structure that is well-known already?
3. Any ideas how to approach this?