neat.py | small implementation of NEAT | Machine Learning library

I'll try to attack the main issues. My understanding of your actual problem is quite limited as I don't know what your code actually does.

This doesn't work as pygame is built on SDL2 which states "You should not expect to be able to create a window, render, or receive events on any thread other than the main one."

Same as above, the display only work in the main thread.

You could pickle the methods using something like dill, but it fells (to me) wrong to copy full on python object between processes. I'd go for another solution.

You need to use to serialize Python objects in order to send them between processes.

I don't understand what this mean.

It seems to me that the problem could be solved with better separation of data and visualization. The training should have no knowledge about any visualization, as it's not dependent on how you want to display it. So there should not be any reason to share the pygame display.

Once this is done, it shouldn't be too much problem (multi-threading always causes problems) to do what you're trying to do. Regarding the pickle issue; I'd try to avoid pickling Python objects and functions, and instead just pass basic primitives between threads and processes. It seems like you should be able to assign self.fitnessFromArray with a simple int instead and based on its value do the min/avg/max calculation in the thread/process instead.

If you want to do threading, then the main thread will take care of the rendering. It'll also spawn threads for the training. When the threads are completed they'll return their result (or put it in a thread safe storage) and the main thread will poll the data and display the result. If the work done by the training takes longer than one frame, then divide up the work so each thread only partially train and can continue where it left off the next frame.

The principal is the same if you instead want separate processes. The main process starts up several training processes and connect to them via sockets. From the sockets, you'd poll information about the state of the program and display it. It would basically be a client-server architecture (albeit on localhost) where the training scripts are servers and the main process is a client.