Many PDEs modeling nature are derived by reduced-order models of many-body descriptions, most famously the Navier-Stokes equations from the Chapman-Enskog expansion of the Boltzmann equation as well as related notions in stochastic processes, e.g. the relationship between the Langevin and Fokker-Planck equations. Moreover, in statistical mechanics we have the moment method which in general gives us a procedure for coarse-graining by making assumptions on the form of higher statistical moments.
Meanwhile at the macroscopic scale, we have various characterizations of PDE as elliptic, hyperbolic, parabolic, etc. which are statements about the structure of the equations themselves (eg. operator polynomials), and rather hard to characterize in general.
In an effort to better understand general PDEs, I'm interested in a reverse question to the one central to statistical mechanics; are there general procedures for determining microscopic descriptions of which a particular macroscopic model is a continuum limit?
Is it possible to obtain a better understanding of continuum solutions by examining corresponding dynamics on a "fine-grained" description?
In physics there is something called universality, which basically means that in many cases, the details of the microscopic model get washed out when we look at the system as a continuum. As an example, think of a random walk on a square lattice or on a hexagonal lattice. In both cases, the continuum behavior looks like diffusion, is symmetric in all directions (so that the symmetry of the lattice disappears) and the only difference, if at all, is in the specific value of the diffusion coefficient. So no, typically there is not a unique way to unmash the potatoes. Having said that, there are ways to extract a microscopic model given a continuum model. As you have observed, it is similar to moving from the Fokker-Planck equation to the Langevin equation: interpret the diffusion term in the equation as coming form a microscopic random walk, the first order derivative as an external force and the "self" terms (not containing derivatives) as interactions between the random walkers.
The main point of universality is that microscopic behaviors with the same symmetry (as you can see, lattice symmetry is not necessarily a relevant one) will map into the same macroscopic behaviors up to the values of the coefficients.