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

Data-Driven Optimal Closures for Mean-Cluster Models: Beyond the Classical Pair Approximation