How can I determine the mathematical relation between the input and output variables? I would like to take in some input values for $n$ variables, say $R$, $B$, and $G$. Let $Y$ denote the response variable of these $n$ inputs (in this example, we have $3$ inputs). Other than these, I would like to use a reference/target value to compare the results. 

Now, suppose the relation between the inputs ($R$, $B$ and $G$) with the output $Y$ is (let's say):

$$Y = R + B + G$$

But the system/machine has no knowledge of this relation. It can only read its inputs, $R$, $B$ and $G$, and the output, $Y$. Also, the system is provided with the reference value, say, $\text{REF} = 30$ (suppose). 

The aim of the machine is to find this relation between its inputs and output(s). For this, I have come across some quite useful material online like this forum query and Approximation by Superpositions of a Sigmoidal Function by G. Cybenko and felt that it were possible. Also, I doubt that Polynomial Regression may be helpful as suggested Here.

One vague approach that comes to my mind is to use a truth table like approach to somehow deduce the effect of the inputs on the output and hence, get a function for it. But neither am I sure how to proceed with it, nor do I trust its credibility. 

Is there any alternative/already existing method to accomplish this?
There are always a large number of possible functions that can produce a given set of input-output values. The challenge is to find a simplest function (according to whatever criteria you choose) to produce those values.

One approach is to write a general function of the input variables, comprising terms of all order in R, G, and B, with a coefficient for each term, then search for values of the coefficients that A) reproduce the known input-output values accurately and B) leave the largest number of coefficients equal to zero.  

Several different algorithms can be used to do the search efficiently.  My choice would be a genetic algorithm to seek the minimum of the RMS difference between the produced and known I-O values, summed with the product of a gradually increasing parameter and the number of nonzero coefficients.