What are some complexity classes for neural networks? Turing machines and neural networks are equivalent in their expressive powers, but as models of computation they are different. Turing machines come pre-configured with their transition functions while the neural networks configure weights to build learnable computable functions using input data.

Is the class of problems efficiently solvable by a neural network equivalent to the class P or NP... or some normal complexity class or do they have their own complexity classes that are not equivalent to normal ones.
Basically, no one has proven anything useful about which tasks a neural networks are able to efficiently learn to solve.

What we can say is: they're not magic pixie dust.  They can't solve NP-complete problems in polynomial time, unless P = NP.

In practice, for most tasks that you'd find in a classical algorithms class, human-made algorithms vastly outperform neural networks.  The situations where neural networks are better tends to be tasks that don't admit a crisp problem statement or specification of the desired behavior.