Could be almost anything, except potato blight
. Look at the brown bits closely, even use a magnifier, and don't forget to look under the leaves too. If the spots have a radiating circular shape then it is some type of fungal or mildew problem, which can be caused by too much or too little water
(probably too little as they like plenty), or not enough ventilation or space around the plants. The spots could be caused by insect activity under the leaves, or by the sun (wot sun?) burning through water drops on the surface. There are also all sorts of rusts and viruses and so on, some of which are non-specific and others are specific to each species. The Readers Digest gardening book says that if you don't know what it is, spray mightily with any and every spray you can lay your hands on
It is definitely a bit outdated but my good books are all in the bedroom and OH is sleeping....... If they were my plants I would pick off and burn the worst affected leaves, spray and feed with a pick-me-up such as liquid seaweed, give them a good drenching if they're dry and hope they last until they give me a crop
Sunflowers, which are related, tend to get brown patches on the leaves too and are not badly affected by them. Jerusalem Artichokes are pretty sturdy things so I wouldn't worry too much.