If you have a large number of correlated response variables, each with its own set of predictors, but you think the relationship of the response to the predictors is the same for all predictors, what neural network architecture should be used? This differs from the case where all responses have the same inputs.
For example, if for 1000 stocks over 5 years you predict monthly returns each month using the 12-month return of each stock and the price/earnings ratio of each stock, you have 5*12 = 60 observations. Each observation has 1000 responses with 2 predictors for each response. Stock returns are positively correlated to each other, so the responses cannot be assumed independent.