Data-driven optimal closures for mean-cluster models: Beyond the classical pair approximation

This study concerns the mean-clustering approach to modeling 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, although it offers a good approximation of the actual time evolution of the cluster concentrations on short timescales only. 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.