  We introduce Coarse-Grained Nonlinear Dynamics, an efficient and universal
parameterization of nonlinear system dynamics based on the Volterra series
expansion. These models require a number of parameters only quasilinear in the
system's memory regardless of the order at which the Volterra expansion is
truncated; this is a superpolynomial reduction in the number of parameters as
the order becomes large. This efficient parameterization is achieved by
coarse-graining parts of the system dynamics that depend on the product of
temporally distant input samples; this is conceptually similar to the
coarse-graining that the fast multipole method uses to achieve $\mathcal{O}(n)$
simulation of n-body dynamics. Our efficient parameterization of nonlinear
dynamics can be used for regularization, leading to Coarse-Grained Nonlinear
System Identification, a technique which requires very little experimental data
to identify accurate nonlinear dynamic models. We demonstrate the properties of
this approach on a simple synthetic problem. We also demonstrate this approach
experimentally, showing that it identifies an accurate model of the nonlinear
voltage to luminosity dynamics of a tungsten filament with less than a second
of experimental data.
