I'll be frank, I'm a bit out of my depth with this. That said, I wonder if there is a heuristic method that takes in a set of measurements of an asymptotic function and, from these measurements, is able of infer a decomposition of the asymptotic function into its constituent components.
For example, imagine we have the following data:
$x=[1,2,3,4,5,6,7,8,9]$
$y=[3,12,35,80,157,280,471,768,1241]$
Where the function is:
$f(x) = 2^x + x^3$
Which, asymptotically, would be:
$O(2^n)$
Are there any methods to take in the function evaluations (i.e. $y$) and infer the components driving the dynamics of the function (i.e. $2^x$ and $x^3$). My understanding is that such a method would necessarily be heuristic because the number of functions that might result in the values of $y$ upon input of $x$ is infinite.
Caveat: I may not have the language to search for such a method, but I haven't had much progress using terms like 'decomposition of asymptotic functions.'