A (convoluted) way to reduce the number of states

[SouthEndMusic]

Say in a model we make a selection of (non UserDemand) connector nodes, such that for each of these connector nodes the connected Basins are connected to at most one of these selected connector nodes. Say we eliminate the state associated with the flow trough the selected connector nodes. Then within the rhs we have to solve for these states.

Say for one of the selected connector nodes $u$ is the implicit singleton state we have to solve for (i.e. the cumulative flow trough the selected connector node). We know the level in the connected basins as a function of $u$, say $h_a(u), h_b(u)$, and the ODE for u:

$$ \frac{\text{d}u}{\text{d}t} = f(h_a(u), h_b(u)) $$

Let's now look at what we have to do in in water_balance!. For each selected connector node, we know the implicit state $u$ at tprev, and we have to know it at the input t. Thus we have to solve the above scalar ODE on the interval [tprev, t] in a differentiable way. I'm quite hopeful that this would work when using ForwardDiff. Note also that the ODEs for the selected connector nodes can be solved in parallel.

[Huite]

Not sure how feasible this is, but:

The primary motivation in decreasing the number of states here is to lower the amount of linear algebra work, right? Smaller systems, faster factorization, etc. It is possible to do this reduction after the fact instead, after the rhs function? (In some sense like a preconditioning step.) I'm mentioning this because it might allow the rhs formulation to remain more intuitive (and not convoluted, as you mention in the title).

(Claude mentions investigating a DAE formulation as well; that doesn't seem too attractive to me as DAEs always require implicit solvers.)