system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: How are pre amps for microphones or guitar amps programmed into digital plug-ins to accurately recreate the tone you'd receive from the actual device?
gpt
The designers create a mathematical model of how the amplifier works in theory. Then they compute that model for your given input on the computer to generate the output. 

With amplifiers the big thing they're after is something called non-linearity. 

Linearity means if A goes in and X comes out, B goes in and Y comes out, than if A + B goes in you'll get X + Y out. 

With an amp, if A goes in you *should* get G×A out, where G is a fixed constant. One of the most common things to happen in an amp is something called "clipping." Clipping happens when G×A is greater than the voltage supply for the components. The amp will "clip" off the top of the signal at the voltage supply level. So if you wanted to model simple clipping you could use a piece wise function

    B = G×A, GA <= V max
    B = Vmax, otherwise 

There's your simple clipper model for a preamp. The issue though is that electronics don't just act non linear depending on supply voltages. In the case of clipping, the signal doesn't get cut right off exactly at the supply. It sort of gets rounded off as the signal gets higher. So to get an accurate sound you need a better model. 

Being able to create a good model isn't trivial. It's based both on circuit analysis and physics as well as theoretical signal processing. There's good money in it so the big companies don't share how they figured it all out.