Train a feed forward neural network indirectly
Your idea with additional layer is good, although the problem is, that your weights in this layer have to be fixed. So in practise, you have to compute the partial derivatives of your R^2->R mapping, which can be used as the error to propagate through your network during training. Unfortunately, this may lead to the well known "vanishing gradient problem" which stopped the development of NN for many years.
In short - you can either manually compute the partial derivatives, and given expected output in R, simply feed the computed "backpropagated" errors to the network looking for R^2->R^2 mapping or as you said - create additional layer, and train it normally, but you will have to make the upper weights constant (which will require some changes in the implementation).
I am faced with this problem:
I have to build an FFNN that has to approximate an unknown function f:R^2 -> R^2. The data in my possession to check the net is a one-dimensional R vector. I know the function g:R^2->R that will map the output of the net into the space of my data. So I would use the neural network as a filter against bias in the data. But I am faced with two problems:
Firstly, how can I train my network in this way?
Secondly, I am thinking about adding an extra hidden layer that maps R^2->R and lets the net train itself to find the correct maps and then remove the extra layer. Would this algorithm be correct? Namely, would the output be the same that I was looking for?