Is there a freely usable map of Earth cave systems and/or density? I am considering creating some sort of model of the Earth using real world collected data. Things like worldwide height maps certainly exist, as do many maps of minerals, rainfall, etc.

Are there worldwide data available on underground cave systems?

By density, I mean the likelihood of there being a cave in a certain place, given coordinates. Hence it does not have to be exact (although this would be better), but a mathematical model or approximation would be useful if the former does not exist.
You might like to check World Cave List which has a pretty extensive list of caves, their depths, and lengths.
For example:

This list has been automatically produced from our World Caves Database.
Total depth and length of all caves currently collected in the database:

Number of caves = 2424
Caves deeper than 300m = 1075
Caves longer than 3kms = 1628
Cumulated depth = 648 932 m
Cumulated length = 17 930 192 m