Data structure for finding nearest squares from current square For my project I can have 0-500 moving squares. The largest possible square width is at most 3 times longer than the smallest possible square width. The squares can move up to 3 of their own body's width per second. About 1 square is added and about 1 square is deleted every second.

What I'm trying to implement now is a neural network to drive the speed and direction of the squares. Each square will have their own neural network which will take the x and y position of the 5 nearest squares to it. To find the nearest squares, I will be taking the euclidean distance from the center of their squares bodies. 

My problem is that calculating the distance of each square to every single other square - to find its 5 nearest other squares - is going to make my program run at a snail's pace. 

So what is a good data structure I could use to reduce the number of checks I'm performing?

Also I heard that kd-trees were good at finding the nearest neighbor. However, because the squares are going to be moving a lot I'm not sure if it'd actually be the best data structure to use.
Since you only measure distances from the centers of the squares, you can restate your problem as follows:
Maintain a dynamic collection $X$ of $n$ points in Euclidean space so that you can quickly find the $k$ nearest neighbors in $X \setminus \{p\}$ of a query point $p \in X$. 

Using the data structure described here you can maintain such a collection in $O(\log^3 n)$ expected amortized time per insertion and $O(\log^6 n)$ expected amortized time per deletion,  after a preproessing requiring $O(n \log^2 n)$ expected time.

The data structure supports nearest neighbor queries (given a point $q$, report the point in $X$ that is closest to $q$) in $O(\log^2 n)$ worst-case time. Then, you can report the $k$ nearest neighbors of $p \in X$ in $O(k \log^6 n)$ expected amortized time as follows:


Delete $p$ from $X$ in $O(\log^6 n)$ expected amortized time.
Repeat $k$ times:


Query the data structure for the nearest neighbor $p' \in X$ of $p$ in $O(\log^2 n)$ worst-case time.
Report $p'$ as one of the $k$ nearest neighbors of $p$.
Delete $p'$ from $X$ in $O(\log^6 n)$ expected amortized time.

Reinsert all deleted points in $X$ in $O(k \log^3 n)$ expected amortized time.