I will have to read these papers then. My intuition is that it's impossible to usefully use PoW to train neural networks because you have to rely on user-submitted training data in order to work which allows you to cheat by pre-determining the solution to your own work.

It's not a terrible idea, but I've yet to see it be inplemented. Gridcoin is one typical example where it's just PoS with "useful PoW" tacked on for token distribution, and doesn't actually use PoW for security.