I considered considering this ridiculously over-complicated approximate solution. Train a neural network to sort with a bunch of examples of sorted arrays.
I mean it might technically be possible if you did some kind of recurrent neural network and gave the inputs through time rather than through space. And you would also have to let it interact with some kind of external memory or workspace.
Perhaps you could train it to just perform operations on the list itself (compare two elements, swap two elements, etc.) That would be the simplest way.