How can I go about building a model for large number of outputs? I have previously worked on small-scale feedforward neural network problems.

But I have started working on a new project where the goal is to predict air quality in 25 locations throughout the country a day ahead. Now, I am quite well-versed with the air quality side of things.

My question:

In a problem like this, would I develop 25 independent models (which share the same input structure) or one model with 25 outputs. 

I guess what I want to do - is there something like parallel neural networks? Or is this 25 different problems? I have mostly worked on physical models where the physics would be shared by all 25 locations. And the inputs would be different.

Would this be a data parallelism or a model parallelism problem?
Neural networks can have 25 outputs.  You would probably get a slightly more accurate result with 25 independent models, but the computation and training time would be 25x of one independent model.  One model with 25 outputs would only take slightly longer than one independent model.