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By observing some system list of inputs and outputs is collected. Is there a general algorithm in mathematics for finding (approximate) polynomial function that will map inputs to outputs? It would be great if the procedure can be generalized to more than one variable.

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What is the motivation for this problem? In particular, are you looking for a good model or an exact fit? These are not generally the same. –  smackcrane Nov 10 '11 at 19:59
    
I guess this question is related to artificial intelligence. Also, as I know there are computer programs which evaluate the function by considering the inputs and outputs. –  James Nov 10 '11 at 20:59

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For a polynomial, you can do Lagrange interpolation to any desired order. If you do one to a given order and the next point(s) fit, that would give you good confidence.

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