Equation-free modeling performs coarse-scale calculations using short runs of a finer-scale simulator when an explicit coarse evolution equation is unavailable. A lifting step constructs microscopic states consistent with coarse variables; simulation advances them; restriction measures the resulting coarse state. These operations define a coarse time-stepper. [ref]
Equation-Free, Coarse-Grained Multiscale Computation: Enabling Microscopic Simulators to Perform System-Level Analysis by Yannis G. Kevrekidis and others. 2003, lifting, restriction and coarse time-stepper framework
For example, microscopic cell states can be initialized to match a population density, evolved briefly, and averaged to estimate a coarse change rate. That estimate can feed a longer projective step or a steady-state calculation. The method needs suitable coarse variables and separation between fast relaxation and slow evolution. If unresolved memory remains important, two microscopic preparations with the same coarse value can evolve differently, invalidating the assumed closure.
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Equation-free modeling is a computational approach used in scientific research, particularly in complex systems, where the underlying equations governing the dynamics of the system are either unknown, too complex to solve analytically, or too costly to simulate directly. The focus of equation-free modeling is on the system's emergent behavior rather than on deriving explicit equations that dictate that behavior.