Black walnut toxicity
Most vegetable plants are adversely affected by juglone, the compound produced by black walnuts. I was not able to find anything on sweet potatoes in particular just that some crops, like carrots, can tolerate this compound to some extent. It's not just the chemical that can affect the plants but often if vegetables or flowers are grown near trees they don't get enough light and the tree roots compete for water and nutrients. If you have a spot that gets 6 to 8 hours of sun each day, my suggestion would be to try growing a few sweet potatoes and see how they do. You might have to water them more during dry periods if tree roots are also in the area. So I would try a small experiment but I anticipate that sweet potatoes would not do well if grown near a black walnut.