Which of the following sentences is correct?
- The numbers are experimentally determined.
- The numbers are determined experimentally.
Are both (not) correct and if only one of them is correct, what is the underlying rule?
There's no significant difference. It's just a matter of style.
LATER: OP's particular word-pair doesn't occur often enough to easily establish usage figures for the two alternative orders, but I think this NGram shows a clear tendency to write arbitrarily chosen rather than chosen arbitrarily (a grammatically and semantically similar example).
On no authority but my own, I suggest the choice of word order may sometimes be influenced by a tendency to write the contextually more significant word first. In my particular example, for instance, I believe the arbitrary nature of the choice is likely to be more in the writer's mind than the fact of a 'choice' being made at all.