Hopfield networks
Hopfield showed that a network with symmetric feedback connections has an energy that only decreases, so the system settles into stored states.
Why it matters
Neural networks acquired a rigorous physical description, and that brought physicists into the field along with their apparatus.
The network works as an associative memory: given part of a pattern it restores the whole. The proof rests on an analogy with spin glasses and a Lyapunov function. Capacity is limited to roughly 0.14 of the number of units. The paper appeared in years when few were interested in networks, and it is what made the subject respectable again; Hopfield received the 2024 Nobel Prize in Physics for this line of work.