I've often wondered what a theory of computation based in differential equations would look like. We desperately need one. Control theory gives us some tools, but not nearly powerful enough ones.
It's fairly clear by now that AI, at least as applied to domains immersed in the natural world, such as robotics and machine vision, is better thought of as analog in nature. Ten years ago I'd go to robotics conferences and it would not be unusual for me to be the only person in the room who knew that control theory was a thing. Now it's nearly taken over the field. But the kinds of developments we see in the field, both in terms of controls and in terms of machine learning, are essentially experimental in nature. We try some stuff, and see what works. Mathematical proofs, or even a mathematical understanding, of these algorithms comes later, if at all.
To this day we have intuition about why deep networks work, but backing that intuition up with rigorous mathematical analysis has been very challenging and is quite incomplete. But when it works, it works far better than ideas rooted in automata. It would be a tremendous benefit to the field if we could somehow merge the practical power of differential equations and function approximators with the theoretical power of the theory of computation.