I have an unknown function $x \to f(x)$, $f:\mathbb{R}^4 \to \mathbb{R}^4$. All elements $x_i$ are within a closed interval. Every function evaluation is quite expensive. How can I made a statement about the "injectivity" of my function in terms of $f(x) \neq f(y)$, given $x\neq y$ with a limited number of function evaluations?
Edit: Given the nature of the function (a production engineering simulation) and a glance at sample data, I assume $f$ to be both continuous and differentiable.
Thoughts too long for a comment.
I suspect there's no good answer to your question.
How would you begin to think about it for maps from $\mathbb{R}^2 \to \mathbb{R}^2$? if you imagine those as a complex function of a complex variable asking about injectivity by sampling a few values is already hard.
The more you know or expect about the function your simulation is modeling the better the possible answers. Can you run a crude/fast version of your simulation to find (approximate) values at points on a grid in the domain? necessary condition for injectivity is that the function be monotone on each line segment through the domain parallel to the coordinate axes. Then look at various graphical views of the data to get some idea of the shape of your function. Do you expect it to be injective, or not?
you might be able to find out that your function is not in