How Do Artificial Neural Networks Use Output Data to Approximate Functions?

[phoenixthoth]

the ouput nodes are a threshold function of a linear combination of inputs from the previous layer; in a single layered artificial neural network, the output is a linear combination of the input and if the ann has a n_1-n_2-...-n_j-n_(j+1) structure then the n_(j+1) output layers are all a threshold function of a linear combination of the outputs geerated from layer j which has n_j nodes. what is done with the outputs to obtain an approximation to a function?

say you have three output nodes and after you apply the threshold function, let's say it's sigmoid in this case, you get three outputs 0.2, 0.3, and 0.9. what do you do with those numbers if you're trying to approximate a function g?

or let's say you have one ouput node and after you apply the threshold function f to the dot product of the current weight vector and the reults of the previous layer, and you get 0.2. what do you do with that output to approximate a function?

or in general, how would you approximate the function x^2 using an ann? say on the interval [0,1] or [-1,1]...

in all the references I've seen, they go into depth about the error functions, back propogation and how to update the weights, blah blah blah but I'm failing to see where they explain how exactly one uses an ann to fit data.

my semi-ultimate goal would be to use an ann to approximate the fractional iterates of a function...

 

[Hurkyl]

Generally, your output layer has one node for each function you are trying to approximate. The output of that node is your approximation.

 

[phoenixthoth]

the output node is the result of applying a threshold function to a linear combination of outputs of the previous layer or, in the case of no hidden layers, the inputs. however, the threshold function has a limited range such as (0,1) or even {0,1}. so how does one get from the output given by an output node to something suitable to approximate a function with a different range? like if i wanted to approximate Sin[x]+30, how would i actually use the output of a threshold function to do this?