Also not familiar with exactly what 'natural' farming is, but
permaculture stresses observing nature and using it. The One
Straw Revolution by Fukokua is one thing that comes to mind when thinking about 'natural' farming, and that book is rather popular among permies.
Usually, when I describe permaculture to people in agriculture, they say 'Oh, agroforestry!" But I think most conventional agroforestry is more geared to monoculture and it often ignores some things that permaculture finds important, like linking up different species to perform different ecological functions, using multiple canopies/trophic levels, using place appropriate solutions, etc.
Also, permaculture has a strong social and ethical component that could be part of any system of agriculture, but it is only in permaculture that this system is consistently taught.