Nope.
S4 official ROMs are tailor made and tested as a complete software package by Samsung specifically for just one particular S4 model, and it can't be used for other models no matter how similarly sounding they are like your S4 Mini.
They aren't PCs where you can just throw Windows at any random PC, download hardware drivers yourself and expect it to work more or less flawlessly. Android doesn't work that way and it's a key weakness of the platform.
Only thing you can pray for is some random custom ROM dev out there to support Lollipop on your exact model of the S4 Mini but in most likelihood either it doesn't exist or it will be a unusable steaming pile of buggy crap because historically non flagships don't get much attention from the custom ROM scene.