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This work investigates the origins of cation ordering of double perovskitesusing first-principles theory computations combined with machine learning (ML)and causal relations. We have considered various oxidation states of A, A', B,and B' from the family of transition metal ions to construct a diversecompositional space. A conventional framework employing traditional MLclassification algorithms such as Random Forest (RF) coupled with appropriatefeatures including geometry-driven and key structural modes leads to highlyaccurate prediction (~98%) of A-site cation ordering. We have evaluated theaccuracy of ML models by entailing analyses of decision paths, assignments ofprobabilistic confidence bound, and finally introducing a direct non-Gaussianacyclic structural equation model to investigate causality. Our study suggeststhat the structural modes are the most important features for classifyinglayered, columnar and rock-salt ordering. For clear layered ordering, thecharge difference between the A and A' is the most important feature which inturn depends on the B, B' charge separation. Based on the outputs from MLmodels, we have designed functional forms with these features to derive energydifferences forming clear layered ordering. The trilinear coupling betweentilt, rotation, and A-site antiferroelectric displacement in Landau free-energyexpansion becomes the necessary condition behind the formation of A-site cationordering.

Insights into cation ordering of double perovskite oxides from machine learning and causal relations