I have not done this in actual practice, but we can apply theory in a thought experiment.
Fact 1: potatoes grow vigorously and will break open pots that are too small
Fact 2: if the potato plant is happy then it will bear heavy tubers
Fact 3: new potatoes grow from shoots sent out by the parent above the seed potato, that is, from the vegetative shoot, not the feeder roots
Method: So we get a tank and arrange some kind of a grid under the top surface of the tank for the parent tubers to rest on. We allow the tubers to make lanky sprouts so that we can have the tops in the light and the tubers right down on the grid to start. The parent tubers will need support, agreed? The grid will be several inches below the tank lid to allow space for the new tubers to form in the dark. Probably the grid would need to be fine, otherwise the new potatoes will just drop through the grid when small and grow big under the grid without support. But this means a problem; the finer the grid, the harder it will be for a spray to reach the roots above the grid.
So the roots dangle down into the fertilizer mix or aeroponic spray. They have all the food and water they want and do well. This will make for crowding in the opening through the tank lid. So we need a collar which will allow for new vegetative shoots to emerge into the light as required. There have been efforts to grow beet tops hydroponically, and one of the problems is management of the neck of the plant as it, inevitably, gets big. So likely there will be issues at the growing neck.
Also, would you like to risk growing them without the supporting grid? Over to you.