  This study concerns the mean-clustering approach to modelling the evolution
of lattice dynamics. Instead of tracking the state of individual lattice sites,
this approach describes the time evolution of the concentrations of different
cluster types. It leads to an infinite hierarchy of ordinary differential
equations which must be closed by truncation using a so-called closure
condition. This condition approximates the concentrations of higher-order
clusters in terms of the concentrations of lower-order ones. The pair
approximation is the most common form of closure. Here, we consider its
generalization, termed the "optimal approximation", which we calibrate using a
robust data-driven strategy. To fix attention, we focus on a recently proposed
structured lattice model for a nickel-based oxide, similar to that used as
cathode material in modern commercial Li-ion batteries. The form of the
obtained optimal approximation allows us to deduce a simple sparse closure
model. In addition to being more accurate than the classical pair
approximation, this ``sparse approximation'' is also physically interpretable
which allows us to a posteriori refine the hypotheses underlying construction
of this class of closure models. Moreover, the mean-cluster model closed with
this sparse approximation is linear and hence analytically solvable such that
its parametrization is straightforward. On the other hand, parametrization of
the mean-cluster model closed with the pair approximation is shown to lead to
an ill-posed inverse problem.
