No, you shouldn't trim them, at least in theory. The more foliage, the more energy they have to put into the sweet potatoes.

Now, in the real world, probably a little light trimming or heading back won't make a big difference. It will depend somewhat on what stage of development they're in as well -- if the tubers are already large and mostly developed, all some light trimming would do is probably slow down maturity a little tiny bit.

From personal experience, heavy trimming (in my case, by deer) will totally destroy the crop, yielding nothing.

I know it won't help you this year, but in future years, if you have limited space and it's a problem, try growing 'Bush Porto Rico' or 'Vardenman' both of which are "bush" type plants, a lot more compact than many types.

I agree with denninmi.Leaves convert the Solar energy into plant food, without leaves plant will starve.
What next thing should be done to the rapidly growing vines is turn them over to stop from rooting.Growing vines root themselves that consume the food before it will reach to the tubers.Once you find the growing tubers cracking the soil surrounding the stem reduce watering.

Can you train the vines to go up over a trellis, if you don't want them sprawling out of the bed into the paths. You can make a quick trellis out of a hoop of concrete reinforcing wire.

I also read that , besides not pruning - to not allow the vines to take root elsewhere. This may diminish the root crop at the main growing point.