What do you mean by "discrete problems"? By my understanding, weather prediction is basically solving a large system of partial differential equations. Sure the grid methods are a discrete approximation of the true problem, but it is not what I would normally think of as a "discrete problem".
Training a neural network to map "current state of atmosphere" to "state of atmosphere in the future" is definitely possible to do with a neural network, and sounds like a good idea to me.
Training a neural network to map "current state of atmosphere" to "state of atmosphere in the future" is definitely possible to do with a neural network, and sounds like a good idea to me.