Sparse expansions of multicomponent oxide configuration energy using coherency and redundancy

Compressed sensing has become a widely accepted paradigm to construct high dimensional cluster expansion models used for statistical mechanical studies of atomic configuration in complex multicomponent crystalline materials. However, strict sampling requirements necessary to obtain minimal coherence measurements for compressed sensing to guarantee accurate estimation of model parameters are difficult and in some cases impossible to satisfy due to the inability of physical systems to access certain configurations. Nevertheless, the dependence of energy on atomic configuration can still be adequately learned without these strict requirements by using compressed sensing by way of coherent measurements using redundant function sets known as frames. We develop a particular frame constructed from the union of all occupancy-based cluster expansion basis sets. We illustrate how using this highly redundant frame yields sparse expansions of the configuration energy of complex oxide materials that are competitive and often surpass the prediction accuracy and sparsity of models obtained from standard cluster expansions.

Figure 1

Schematic of different domains involved in compressed sensing. CS with redundancy seeks to recover function $H$ in the function domain in the center. Adapted from Candes et al. [23].

Figure 2

Function space representations over the configurations of a single binary site and a symmetric binary diatomic molecule. (a) Function space over a single binary site space. The two different choices for site bases to construct a standard CE are colored red and blue, and each set also includes the purple ${\varphi }_0=1$ function. (b) Function space over symmetrically distinct configurations of the molecule. A CE basis includes either the blue colored or the red colored functions. The generalized Potts frame includes all colored functions (blue/red/yellow). All functions sets also include the magenta colored constant function. The D-RIP for a 2-sparse representation in this case is adapted to the union of all colored planes in (b).

Figure 3

(a) System 2-2: Li-Mn-O-F rocksalt system with binary ($\mathrm{Li}+/\mathrm{Mn}3+$) cation sites and binary (O2-/F-) anion sites. A primitive cell with two sites is used as the formula unit for normalization. (b) System 3-2: Li-Ti-Mn-O-F rocksalt system with ternary ($\mathrm{Li}+/\mathrm{Mn}3+/\mathrm{Ti}4+$) cation sites and binary (O2-/F-) anion sites. A primitive cell with two sites is used as the formula unit for normalization. (c) System 5-3-2: Li-Mn-O-F spinel-like system with quinary ($\mathrm{Li/Mn}2+/\mathrm{Mn}3+/\mathrm{Mn}4+$/vacancy) octahedral cation sites, ternary ($\mathrm{Li/Mn}2+$/vacancy) tetrahedral cation sites, binary (O2-/F-) anion sites. A primitive cell with four sites is used as the formula unit for normalization.

Figure 4

Fit metric statistics for the systems tested using standard CEM with sinusoid site basis, indicator site basis, and generalized Potts frame. The plotted metrics include: cross validation RMSE (CV score), out of sample RMSE (Out RMSE), full data RMSE (full RMSE) for both the training and test structures combined, and the number of nonzero ECI in the fits (sparsity). LiMnOF binary-binary with two sites per formula unit (top), LiMnTiOF ternary-binary with two sites per formula unit (middle), LiMnOF quinary-ternary-binary with four sites per formula unit (bottom).

Figure 5

Sorted fitted coefficient magnitudes (multiplicity times ECI) for sparsest and most accurate model (Full RMSE). LiMnOF binary-binary (top), LiMnTiOF ternary-binary (middle), LiMnOF quinary-ternary-binary (bottom).

Figure 6

Number of nonzero coefficients relative to the sinusoid basis fit for each orbit and norm of coefficients for each orbit for the most accurate models (full RMSE). The vertical dotted lines separate the degree of orbit (pairs/triplets/quadruplets). LiMnOF binary-binary (top), LiMnTiOF ternary-binary (middle), LiMnOF quinary-ternary-binary (bottom).