Suppose you are given a finite list $P_1,\ldots,P_n\in \mathbb{R}^2$ of points and vectors $v_1,\ldots,v_n \in \mathbb{R}^2$. Then you are told that these vectors are the values $F(P_1),\ldots,F(P_n)$ of a parameterized vector field $F:\mathbb{R}^2 \to \mathbb{R}^2$ that associates to each point $P$ of the Euclidean space a vector $F(P) = v_N + v_S$ where
$N,S \in \mathbb{R}^2$ are two unknown points and $M\in\mathbb{R}^+$ is an unknown positive real number
$v_N$ is a vector of magnitude $\frac{M}{d(P,N)}$ and direction from $N$ to $P$ ($d(P,N)$ is the Euclidean distance from $P$ to $N$),
$v_S$ is a vector of magnitude $\frac{M}{d(P,S)}$ and direction from $P$ to $S$
How would you guess the unknown parameters $N,S$ and $M$?
In this simple case I noticed that the magnitude of the vector $F(P)$ increases as $P$ approaches a pole. Therefore it suffices to calculate the gradient of growth on the sampled values and this gives the direction along which one should move to reach the poles.