Were you thinking of fabricating an IC for an individual weather station? Usually they display temperature, humidity, barometric pressure, min/max of all readings, trend up/down. Add an outdoor temp sensor which sends its reading to the indoor unit.
There are internet-connected systems that receive weather forecasts from a central office. (For instance I have one called Weather Direct).
There is large-scale forecasting where the weather service gathers data from many scattered sources, puts it into a simulation program, and observes how the present patterns 'morph' into a future scenario. I believe the software has the featured role in this scale of forecasting, rather than any particular IC.