A Coarse-Graining Procedure for Differential Equation Models

The Renormalization Group framework gives a prescription to determine which macroscopic variables are relevant in a system and sufficient to describe large scale dynamics. Can we develop an analogous methodology to extract important parameters of dynamical systems in general? The Fisher Information Matrix (FIM) formalism measures the response sensitivity of a system to perturbations in its control parameters and has emerged as one way of extracting a hierarchy of relevant variables, even when renormalization isn’t straightforward. The relevant parameter directions are given by eigenvectors of FIM: the hierarchy of relevance is induced by the magnitude of their eigenvalues. This procedure has been done previously for discrete diffusion coarse-grained in time and for the lattice Ising model coarse-grained spatially (Machta et al 2013). Relevant parameters had Fisher information eigenvalues that remained large after microscopic data was discarded. We now extend this analysis to systems of stochastic differential equations utilizing a temporal coarse-graining scheme inspired from Compatible Monte Carlo: specific points in the system are held fixed while the rest of the system is free. Coarse-graining then corresponds to increasing the temporal separation between these fixed points.