This is a page for detailing some of the inner workings to help future contributors to the library.
In the variable "elimination" algorithms, what is actually done is that variables are removed from being states and equations are moved into the
observed category of the system. The
observed equations are explicit algebraic equations which are then substituted out to completely eliminate these variables from the other equations, allowing the system to act as though these variables no longer exist.
However, as a user may have wanted to interact with such variables, for example, plotting their output, these relationships are stored and are then used to generate the
observed equation found in the
SciMLFunction interface, so that
sol[x] lazily reconstructs the observed variable when necessary. In this sense, there is an equivalence between observables and the variable elimination system.