Training a function that maps n-dim to n-dim As an example, say the input is an array of numbers representing an audio snippet and the output is a transformed/filtered version of it.

What would be the proper name for that? Which are examples of algorithms for the job?

EDIT:
More specifically, I want to train audio source separation. The input is a mixed sound (spectrogram) and the output is the sound with some energy removed in certain frequencies. The function needs to recognize some pattern in the input and decide what to remove.
I would call a mapping between N dimensional input and N dimensional output a regression problem. 

If you add more constraints about the relation between the input and output it might be called different names: linear filtering, nonlinear filtering, etc...

some examples on common techniques for that would be: neural networks, regression trees, regularised regressions...