Yes, the shapes can be predicted by numerical solution of the governing PDE. I don't think that's what you had in mind, however. For an arbitrary plate shape,

Yes, the shapes can be predicted by numerical solution of the governing PDE. I don't think that's what you had in mind, however. For an arbitrary plate shape, the solutions do not generally have what you might think of as a nice closed-form expression. For certain specific cases (like a square plate), these modes do have reasonably tidy descriptions.