# How to identify deterministic functions from input-output pairs?

Suppose that we are given a sample of input-output pairs of a deterministic function. To make it concrete, I can generate input-output pairs of a function I myself created (for example $$y = 240 + 20x - 12x^2)$$. The function might be linear or non-linear. But there are no unknown noise or other contributing factors.

The aim is to identify the function from the given inputs and ouputs. In such a situation which techniques are possible to use? Can I use regression techniques or are there better alternatives?

• It depends on what you think the possible functions might be. There will be an infinite number of functions which fit your data. For example, if you have $(1,2), (2,3), (3,5), (4,7), (5,11)$ then OEIS gives many many suggestions including the prime numbers, the number of partitions of $x+1$, $\lfloor (3/2)^{x+1}\rfloor$, and $(3x^4-34x^3+141x^2-206x+144)/24$ Feb 25 at 14:25

I would start by reading up on interpolation and the difficulties at a high level. Then play around with the SciPy tools for interpolation using simple functions you choose.