Suppose an inverse problem with graphical approximation for the system where only a small subset of system features are known hence undetermined scenario. The system can be represented by a graph.
Your goal is to approximate the system in terms of graph-theory where the best approximations for the system are unknown. The parametrisation contains subproblems such as a graph term rewriting problem, where the graphical information are approximated in terms of polynomials to infer system features, and implicitization problems where parametrizations are converted into definining equations for the variety V describing the system, for calculating approximated system features.
This question relates to References on Inverse Problems, Approximation theory and Algebraic geometry in a way that Algebraic geometric approximation and graphical approximation may be related. Now focus on graphical approximation in terms of graphs while best tools still unknown so
Where can you find research, references on inverse problems with graphical approximation?
I know of one inverse problem related to graphs and electrical circuits. Maybe it is of some use.
Calderón's inverse problem asks one to recover the electrical (or heat) conductivity of a body from boundary measurements of current and voltage. In practice the measurements are done by a finite set of electrodes. The usual way of looking at the problem is a partial differential equation $$\nabla \cdot (\sigma \nabla u) = 0,$$ where $u$ is electric potential or temperature, and $\sigma$ is the unknown conductivity, and the body is a subset $\Omega$ of an Euclidian space, or maybe a Riemannian manifold.
The Laplace equation, and (maybe) the conductivity equation (written out above) can also be defined on graphs, which can be used to model electrical circuits.
Consider, then, a graph with some nodes marked as its boundary. On the boundary nodes you know the electrical potential and the current going into or coming out of the graph. You have this information for each solution of the equation (for applied mathematics you only have the data for a finite number of solutions). What can you say about the graph?
Obviously the graph is not unique; you can add a node in the middle of an edge without changing the physical interpretation at all. So you should come up with some sort of minimality condition (much like a regularization parameter in numerical inverse problems) to get uniqueness, I guess.
This discrete approximation has been used to study the continuous Calderón's problem, but I vaguely recall that proving that it converges to the continuous one is difficult.
Liliana Borcea's review article, section 4.5, contains a better introduction and some references: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4950&rep=rep1&type=pdf . The review is not recent, so there might have been progress in the problem since then.