I would expect there is an enormous set of random topologies that could be trained to balance a pole. Indeed, part of the elegant 'magic' of neural nets is that

I would expect there is an enormous set of random topologies that could be trained to balance a pole. Indeed, part of the elegant 'magic' of neural nets is that the topology is fairly irrelevant... the number of layers, the number of nodes, the manner in which they are connected... pretty much any configuration can get you into a mid 90% accuracy on MNIST (and emulating a basic PID algorithm is simpler than MNIST). Of course, I'm referring to basic tasks; clearly topologies matter a great deal with more sophisticated things.