How are various meteorological indices used in weather forecasts calculated? Is there any functional formula?

There is a formula, which is very annoying, and then it is calculated based on the field measured data and substituted into the formula.

In universities studying meteorology, there are always partial differential equations and the like, which is super annoying. Of course, meteorological stations will use computers to make equation models. You only need to input the measured data, and the final calculation may be just A probability may not be accurate, and it may even be scolded by the people. .