# Data-Driven, Physics-Informed Descriptors of Cation Ordering in
Multicomponent Oxides

Jiayu Peng,1 James Damewood,1,2 Rafael Gómez-Bombarelli1,∗

1Department of Materials Science and Engineering, Massachusetts Institute of
Technology,
Cambridge, MA 02139, USA
2Center for Computational Science and Engineering, Massachusetts Institute of
Technology,
Cambridge, MA 02139, USA
∗Correspondence<EMAIL_ADDRESS>

Summary: Data-driven, physically interpretable descriptors of cation ordering
are established for multicomponent perovskite oxides.

> The structural tunability and compositional diversity of multicomponent
> perovskite oxides have enabled their various applications, including
> catalysis and electronics. The cation ordering in these oxides, ranging from
> disordered (i.e., high-entropy) to ordered (e.g., rocksalt), profoundly
> influences their properties. While computational design tools can typically
> predict properties associated with a particular ordering, inferring which
> ordering—if any—will be observed in synthesized oxides remains challenging.
> Here, we leveraged first-principles simulations and machine learning to
> develop data-driven, physics-informed descriptors of experimental ordering
> in multicomponent perovskites and compared them with traditional
> physicochemical descriptors, e.g., ionic radii and oxidation states. The
> fitted low-dimensional classification models correctly rank up to $93\%$ of
> compositions in an experimental dataset of $190$ perovskites between cation-
> ordered and disordered, offering a rigorous benchmark between theory and
> experiments. Furthermore, these descriptors accelerate high-throughput
> virtual screening of multicomponent oxides by predicting their dominant
> ordering to avoid costly, exhaustive simulations of cation arrangements.

## Introduction

Multicomponent perovskite oxides are an essential class of inorganic materials
with diverse key applications, ranging from electronics to renewable energy
(?, ?). Having a general formula of ABO3, these oxides are composed of a
network of B-site cations and oxygen anions, forming corner-sharing BO6
octahedra surrounded by A-site cations. Multiple B-site and A-site cations can
exist in such oxides, and these cations almost span the entire periodic table.
The compositional flexibility of multicomponent perovskite oxides has enabled
their various functions, including high-temperature superconductivity for
electronics (?), colossal magnetoresistance for magnetic devices (?), Li-ion
conductivity for batteries (?), and surface reactivity for heterogeneous
catalysis (?) and electrocatalysis (?, ?).

Concomitant with their compositional versatility, multicomponent perovskite
oxides exhibit variability in the distribution and ordering of different
cations. The specific arrangements of cation sublattices that are periodically
repeated in these oxides can have a profound influence on their physical and
chemical properties. For example, Cu cation layers in cuprate perovskites are
crucial for enabling high-temperature superconductivity (?). Moreover, cation
order-to-disorder tuning in perovskite oxides has been widely leveraged to
optimize their optical absorption (?), magnetic behaviors (?), ion mobility
(?, ?), catalytic activity (?), and electromechanical properties (?),
facilitating their atom-by-atom design for diverse practical applications.

Unfortunately, the design of multicomponent perovskite oxides has been
severely impeded by a lack of physical principles to universally rationalize
and accurately predict their cation ordering. Although a few empirical rules
have been established to correlate the experimental cation ordering in these
oxides with the atomic parameters of their constituent ions (?, ?), such rules
have often been shown to fail across broad oxide space for correctly
classifying perovskite oxides into cation-ordered and disordered structures
(?). For instance, increasing the charge and size differences between B-site
cations in multicomponent perovskite oxides has been reported to favor B-site
sublattices with rocksalt ordering over their cation-disordered counterparts
(?). However, while these correlations can be explained by a simple hypothesis
that the rocksalt ordering is energetically favored to electrostatically
maximize the separation of the more highly charged B-site cations (?), it
cannot quantitatively distinguish cation-ordered perovskite oxides from their
disordered counterparts (?). Similarly, having a large charge mismatch between
A-site ions has been proposed to give rise to A-site layered ordering (?), but
this trend has limited predictive power (?). These observations corroborate
recent discoveries that many traditional empirical principles predicting phase
stability and competition in oxides (?, ?) no longer hold true when re-
examined across a vast compositional space (?, ?). It is thus critical to
rationalize the limitation of such conventional empirical rules and develop
new physics-driven design principles that exhibit greater generalizability and
higher accuracy for inferring the cation ordering in multicomponent perovskite
oxides.

High-throughput atomistic simulations, possibly accelerated with machine
learning, could provide highly accurate, physics-informed design principles of
cation ordering across a wide compositional space of multicomponent
perovskites based on formation energetics (?, ?, ?). Nevertheless, this
exploration of ordering is constrained by a lack of systematic benchmarks
between theory and experiments. Although density functional theory (DFT)
calculations have been shown to broadly and accurately reproduce the
compositional dependence of material properties (?, ?, ?), previous
computational studies on cation ordering have merely evaluated a handful of
oxides by comparing DFT-computed energetics with their experimental cation
arrangements (?, ?, ?, ?, ?, ?, ?). Moreover, DFT-backed, data-driven models
have been designed to infer the formation energetics of oxide structures with
different cation arrangements. However, these models suffer from drawbacks:
they are not physically interpretable since they just map structures to
energies; they are trained with DFT results, lacking comprehensive
experimental validation; and they require relatively large training datasets,
barring the incorporation of the much more limited experimental data (?, ?, ?,
?). Thus, in order to establish descriptors of cation ordering, it is crucial
to systematically examine whether DFT can capture experimental cation ordering
in a large compositional space of multicomponent oxides and compare or combine
these DFT energetics with descriptor-powered, interpretable machine learning
models.

The dearth of cation ordering descriptors has further hindered the high-
throughput virtual screening (HTVS) of multicomponent perovskite oxides. In
HTVS, it has been particularly challenging to representatively, yet
efficiently navigate the immense configurational space of cation ordering.
Currently, the HTVS of such complex oxides is computationally inefficient,
requiring the costly DFT sampling of large supercells (?, ?) with randomly
selected (?, ?) or manually predefined cation arrangements (?, ?). These
existing sampling strategies are not exhaustive, so they lower the otherwise
prohibitive cost of extensively calculating all possible configurations with
DFT. However, it is unclear whether such strategies systematically capture the
low-energy cation arrangements of multicomponent oxides or, more importantly,
whether they reflect experimentally observed ordering. Therefore, it is
imperative to use cation ordering descriptors to boost HTVS by optimizing the
balance between the breadth and cost of DFT sampling in exploring the vast
configurational space of cation arrangements.

Here, combining high-throughput DFT calculations and machine learning, we
developed data-driven, physics-informed descriptors of cation ordering to
universally and accurately infer experimental cation ordering in
multicomponent perovskite oxides. By constructing a dataset of $190$
experimental double perovskites with symmetrically inequivalent cation
arrangements, we showed that these new descriptors significantly outperform
their traditional counterparts, e.g., the charge and size differences of
B-site ions (?, ?, ?), correctly ranking up to $93\%$ oxide compositions
between cation-ordered and disordered. Such descriptors can be rationalized as
physically interpretable parameters, including the thermodynamic likelihood of
each ordering and configurational entropy, which can be obtained from the non-
exhaustive DFT calculations of only a few simple model cation arrangements.
These results use abundant prior experiments to offer a rigorous validation of
our simulation and data science strategies. By coupling these descriptors with
HTVS, we demonstrated that descriptor-driven DFT sampling strategies reside at
the Pareto front of accuracy–cost trade-offs when navigating the
configurational space of cation arrangements, with comparable computational
cost but lower sampling error than their descriptor-free counterparts. This
work highlights an effective paradigm for understanding and predicting
chemical and structural orders in multicomponent materials to accelerate their
data-driven discoveries across a massive compositional space.

## Results

### Building physics-informed descriptors to infer cation ordering

To enable a systematic benchmark between theory and experiments and develop
physics-driven descriptors of cation ordering, we constructed a
compositionally diverse DFT dataset of $190$ experimental multicomponent
perovskite oxides (Figs. 1 and S1; Materials and Methods). In this dataset, we
focused on A2B’B”O6 double perovskites to establish and evaluate descriptors,
as A2B’B”O6 compositions have dominated the literature with experimentally
quantified cation ordering (?). Moreover, we examined selected multicomponent
perovskites with less common compositions and orderings (?, ?, ?, ?, ?) in
detail (Table 1) as an additional evaluation set to further assess the
accuracy and generalizability of these descriptors.

Fig. 1:  Formation energetics of model cation arrangements. (A) Schematic
structures of cation-ordered and disordered multicomponent perovskite oxides,
represented by $8\times 8\times 8$ supercells. (B) Model structures of six
symmetrically inequivalent cation arrangements in a $2\times 2\times 2$
supercell of A2B’B”O6 perovskites. $\Omega$ is the degeneracy of cation
arrangements. Color scheme: A, grey; B’, brown; B”, cyan; O, white. (C) DFT-
computed formation energies of the six model structures relative to that of
the computationally ground-state cation arrangement for an experimental
dataset of $190$ A2B’B”O6 perovskites.

With the datasets, we proposed physically interpretable descriptors of cation
ordering based on the formation energetics of simple model cation
arrangements. While large supercells have been widely utilized to capture the
quasirandom cation sublattices of multicomponent oxides (Fig. 1A), these
massive supercells can be too computationally expensive for high-throughput
investigation (?). To tackle this limitation, we examined the formation
energetics of simple, less expensive $2\times 2\times 2$ supercells with
symmetrically distinct cation arrangements, offering a computationally
efficient approach for navigating the energy landscape of cation ordering in
complex perovskites. Specifically, distributing four B’ and four B” cations
among the eight B sites in $2\times 2\times 2$ A2B’B”O6 supercells leads to as
many as $(_{4}^{8})=70$ combinations, but such combinations can be reduced to
only six symmetrically inequivalent configurations (Fig. 1B). Among the six
configurations, three of them have been named layered, columnar, and rocksalt
arrangements, with cations alternating in one, two, and three dimensions,
respectively (?, ?, ?). Computationally, the rocksalt arrangement was found to
be the ground-state configuration for most of these $190$ oxides (Fig. 1C),
consistent with prior experimental observations that, under ambient pressure,
all but one known cation-ordered A2B’B”O6 oxides have B-site sublattices with
rocksalt ordering (?, ?). Although such a rocksalt arrangement tends to be the
ground-state configuration, other cation arrangements can also be
thermodynamically favorable if their formation energies are only slightly
higher than their ground-state counterpart, especially if they have favorable
configurational entropy. Thus, the tendency to deviate from randomness may be
characterized by the energy gap between ground-state and higher-energy cation
arrangements ($\Delta E$) or, more rigorously, the thermodynamic likelihood of
forming a configuration $i$:

$P_{i}=\Omega_{i}\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}},\quad
i=1,\ldots,N_{\mathrm{unique}}$ (1)

where $E_{i}$ is the formation energy of the cation configuration $i$ (Fig.
1C), $\Omega_{i}$ is the degeneracy of this configuration (Fig. 1B),
$N_{\mathrm{unique}}$ is the number of symmetrically unique configurations
($N_{\mathrm{unique}}=6$ for A2B’B”O6), $k_{\mathrm{B}}$ is the Boltzmann
constant, and $T$ is the temperature. Another parameter that potentially
captures the general degree of order or disorder is the configurational
entropy, which weights across all possible orderings:

$S_{\mathrm{conf}}=-\sum_{i=1}^{n}\frac{\Omega_{i}}{\ln{\sum_{k=1}^{N_{\mathrm{unique}}}\Omega_{k}}}\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}\ln{\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}}$
(2)

which has a range between $S_{\mathrm{residual}}$ and $1$, where
$S_{\mathrm{residual}}$ is the residual entropy stemming from the
configurational multiplicity of the symmetrically unique ground-state
arrangement (Eq. 6; Materials and Methods). Intuitively, the difference
between $S_{\mathrm{residual}}$ and $S_{\mathrm{conf}}$ quantifies whether
low-lying alternative orderings to the ground state are expected to be
populated, beyond the sheer configurational entropy of the degenerate
arrangements of the ground state ($S_{\mathrm{residual}}$). Overall, these
three parameters, $\Delta E$, $P_{i}$, and $S_{\mathrm{conf}}$, characterize
the statistical thermodynamics of various cation configurations in
multicomponent oxides and can thus serve as physics-driven descriptors to
infer their experimental cation ordering.

### Evaluating cation ordering descriptors with experimental datasets

We compared these physically interpretable thermodynamic parameters with
traditional atomic descriptors (?, ?, ?) to investigate the performance in
classifying multicomponent perovskite oxides as cation-ordered or disordered
across a wide oxide space. Specifically, for the dataset of $190$ A2B’B”O6
oxides (Fig. 1), we established one- or multi-dimensional descriptors based on
the combination of these parameters and fitted a logistic classifier to
predict whether such oxides are experimentally cation-disordered or ordered
with B-site rocksalt ordering (Fig. 2; Materials and Methods). The descriptor
performance was assessed by the confusion matrices and the receiver operating
characteristic (ROC) curves. While a confusion matrix characterizes the
classification accuracy at the optimized decision boundary, the area under the
curve (AUC) in ROC curves quantifies the overall accuracy at various
classification thresholds, providing a comprehensive, systematic evaluation of
the descriptor performance, and is a more preferred metric for imbalanced
classes. Intuitively, the ROC–AUC of a descriptor can be understood as the
average accuracy (or probability) that this descriptor ranks an experimentally
cation-ordered perovskite oxide as higher-likelihood of being ordered than a
cation-disordered one.

Increasing $S_{\mathrm{conf}}$ and lowering the likelihood of forming the
rocksalt cation configuration ($P_{\mathrm{r}}$) were found to markedly
outperform traditional atomic descriptors in favoring disordered B-site
sublattices over cation-ordered ones. Specifically, while decreasing the
charge ($\Delta n_{\mathrm{ox(B)}}$) and size differences between B-site ions
($\Delta r_{\mathrm{ion(B)}}$) has been widely reported to lead to B-site
sublattices with cation disorder (?, ?, ?), these two parameters can hardly
effectively classify experimentally cation-ordered and disordered oxides.
$\Delta n_{\mathrm{ox(B)}}$ and $\Delta r_{\mathrm{ion(B)}}$ have ROC–AUC
scores of $0.78$ and $0.69$, respectively, and coupling them as a two-
dimensional (2D) descriptor results in an AUC of $0.81$ (Fig. 2A), indicating
an $81\%$ accuracy in ranking cation-ordered oxides above their disordered
counterparts. Nevertheless, although this 2D descriptor behaves better than
other atomic parameters such as the electronegativity difference
($\Delta\chi_{\mathrm{(B)}}$), which has a ROC–AUC comparable to a trivial
random classifier (Fig. 2A), it cannot quantitatively differ cation-disordered
perovskites from ordered oxides in its 2D parameter space (Fig. 2D). The
limitation of $\Delta n_{\mathrm{ox(B)}}$ and $\Delta r_{\mathrm{ion(B)}}$ is
corroborated by their confusion matrices, where more than half of
experimentally cation-disordered perovskites are misclassified as ordered
(Fig. S2). In contrast, thermodynamic parameters, including $P_{\mathrm{r}}$
(Eq. 1) and $S_{\mathrm{conf}}$ (Eq. 2), lead to a 2D descriptor that has an
AUC of $0.86$ (Fig. 2B) and can effectively distinguish cation-ordered oxides
from their disordered counterparts (Fig. 2E), highlighting the power of
thermodynamic descriptors in understanding and predicting experimental cation
ordering.

Fig. 2:  Descriptor performance for classifying A2B’B”O6 perovskites as
cation-ordered or disordered. (A to C) Performance of one- and multi-
dimensional descriptors using logistic regression with five-fold cross-
validation, quantified by ROC curves with AUC in parentheses. Dashed lines
show the ROC curves of a random classifier (AUC $=0.5$).
$\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and $\Delta
n_{\mathrm{ox(B)}}$ are the differences between the electronegativities, ionic
radii, and oxidation states of B-site ions, respectively. $\Delta
E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$, and $P_{\mathrm{r}}$ are the energy
gap between the layer and rocksalt arrangements, the configuration entropy
(Eq. 2), and the likelihood of generating the rocksalt configuration (Eq. 1),
respectively. (D to F) Decision boundaries of top-performing multi-dimensional
descriptors. $P_{\mathrm{ordered}}$ is the predicted likelihood that an oxide
should be classified as cation-ordered through logistic regression.

Hybridizing thermodynamic and atomic parameters further contributes to highly
predictive descriptors of cation ordering. Particularly,
$\\{S_{\mathrm{conf}},P_{\mathrm{r}},\Delta r_{\mathrm{ion(B)}},\Delta
n_{\mathrm{ox(B)}}\\}$ serves as an effective four-dimensional (4D) descriptor
with a ROC–AUC of $0.90$ (Fig. 2C). Moreover, replacing $S_{\mathrm{conf}}$
and $P_{\mathrm{r}}$ with the energy gap between the layer and rocksalt
configurations ($\Delta E_{\mathrm{l,r}}$, Fig. 1) generates a comparably
accurate three-dimensional (3D) descriptor with an AUC of $0.89$. The
comparable performance of such hybrid descriptors can be explained by the
strong influence of $\Delta E_{\mathrm{l,r}}$ on $S_{\mathrm{conf}}$ and
$P_{\mathrm{r}}$. Moreover, the higher accuracy of
$\\{S_{\mathrm{conf}},P_{\mathrm{r}},\Delta r_{\mathrm{ion(B)}},\Delta
n_{\mathrm{ox(B)}}\\}$ and $\\{\Delta E_{\mathrm{l,r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$ than thermodynamic or atomic
descriptors separately can be justified by the orthogonality among these
descriptors (Fig. S3). Intriguingly, when training the logistic regression
model with $\\{S_{\mathrm{conf}},P_{\mathrm{r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$ or $\\{\Delta
E_{\mathrm{l,r}},\Delta r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$, we
did not provide the model with the information that eight out of the $190$
A2B’B”O6 oxides are partially cation-ordered, but such descriptors appear to
capture the experimentally partial cation ordering in these oxides by having
their data close to the decision boundaries (Figs. 2F and S2K). Given such an
experimental ambiguity, filtering out these eight experimentally partially
cation-ordered perovskites can result in ROC–AUC scores of $0.93$ and $0.92$
for $\\{S_{\mathrm{conf}},P_{\mathrm{r}},\Delta r_{\mathrm{ion(B)}},\Delta
n_{\mathrm{ox(B)}}\\}$ and $\\{\Delta E_{\mathrm{l,r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$ (Fig. S4), respectively,
demonstrating that such 4D and 3D hybrid descriptors have up to $93\%$ and
$92\%$ accuracies in ranking cation-ordered A2B’B”O6 perovskites above their
disordered counterparts for unambiguous experimental data.

Table 1:  Descriptor performance for perovskite oxides with exotic cation
orderings.

Composition | Experimental ordering | $N_{\mathrm{total}},\ N_{\mathrm{unique}}$ | $P_{\mathrm{exp}}$ | $S_{\mathrm{conf}},\ S_{\mathrm{residual}}$
---|---|---|---|---
Nd2CuSnO6 | B-site layered (?) | $70,\ 6$ | $0.98$ | $0.46,\ 0.42$
Pr2CuSnO6 | B-site layered (?) | $70,\ 6$ | $1.00$ | $0.42,\ 0.42$
Sm2CuSnO6 | B-site layered (?) | $70,\ 6$ | $0.98$ | $0.45,\ 0.42$
CaMnTi2O6 | A-site columnar (?) | $70,\ 6$ | $1.00$ | $0.43,\ 0.42$
CaCu3Fe4O12 | A-site body-centered (?) | $28,\ 3$ | $1.00$ | $0.42,\ 0.42$
LaCu3Fe4O12 | A-site body-centered (?) | $28,\ 3$ | $1.00$ | $0.42,\ 0.42$
CaCu3Cr2Sb2O12 |  | B-site rocksalt (?)
---
$1960,\ 26$ | $0.37$ | $0.60,\ 0.27$
| A-site body-centered (?)
---
$0.00$

  * $N_{\mathrm{total}}$ is the number of all possible cation arrangements in a $2\times 2\times 2$ supercell, while $N_{\mathrm{unique}}$ is the number of symmetrically unique cation configurations. $P_{\mathrm{exp}}$ is the likelihood of forming the arrangement equivalent to experimental ordering (Eq. 1). $S_{\mathrm{conf}}$ is the configurational entropy (Eq. 2) with a range between $S_{\mathrm{residual}}$ and $1$, where $S_{\mathrm{residual}}$ is the residual entropy (Eq. 6). Nd2CuSnO6, Pr2CuSnO6, and Sm2CuSnO6 have only been synthesized under high pressure (?).

These physics-informed thermodynamic descriptors, e.g., $S_{\mathrm{conf}}$
and $P_{\mathrm{r}}$, can be generalized beyond the binary classification of
A2B’B”O6 into cation-ordered perovskites with rocksalt ordering or disordered
ones (Table 1). For the additional evaluation set of oxides with either B-site
layered ordering (?) or A-site columnar (?) or body-centered ordering (?, ?),
to the best of our knowledge, there is a lack of atomic descriptors for
rationalizing such exotic cation orderings. However, our thermodynamic
descriptors suggest that the experimentally observed orderings for these
oxides tend to be the ones with the highest thermodynamic likelihood of
forming the configuration equivalent to experimental ordering
($P_{\mathrm{exp}}$). We also found there are negligible differences between
$S_{\mathrm{conf}}$ and $S_{\mathrm{residual}}$, suggesting single dominant
ordered phases, because a large difference would indicate that there are
alternative orderings with a significant population to lead to disorder. These
results highlight the power of $S_{\mathrm{conf}}$ and $P_{\mathrm{exp}}$ in
predicting uncommon cation orderings. Nevertheless, extending the
compositional space to perovskites with cation orderings in both A and B sites
makes it difficult to find maximal and minimal values for $P_{\mathrm{exp}}$
and $S_{\mathrm{conf}}$, respectively. For example, for CaCu3Cr2Sb2O12 with
experimentally observed B-site rocksalt and A-site body-centered orderings
(?), the $P_{\mathrm{exp}}$ for such B-site and A-site orderings are $0.37$
and $0.00$, respectively. Moreover, the $S_{\mathrm{conf}}$ ($0.60$) of
CaCu3Cr2Sb2O12 is much higher than its $S_{\mathrm{residual}}$ ($0.27$). Such
a mismatch between simulations and experiments can be potentially explained by
the too-high configurational multiplicity of cation arrangements (i.e., $1960$
possible cation arrangements and $26$ symmetrically unique cation
configurations in a $2\times 2\times 2$ supercell), as the energy gap between
two symmetrically distinct configurations can be too small relative to the
uncertainty in DFT (${\sim}20\ \mathrm{meV}$ per atom, (?)). However, these
energetically comparable configurations are hypothesized to have a minimal
difference in their arrangements and properties. Thus, evaluating either of
such arrangements might only result in a negligible influence on the
utilization of cation ordering descriptors in investigating such complex
perovskites. Overall, we confirmed the generalizability of cation ordering
descriptors in understanding the various types of cation ordering across a
broad compositional space.

### Assessing the robustness and expressivity of cation ordering descriptors

Fig. 3:  Sensitivity analysis of thermodynamic and atomic descriptors. (A to
C) Statistics of selected DFT-computed properties for the experimental dataset
of $190$ A2B’B”O6 oxides, including the minimum energy above the convex hull,
the maximum cosine distance between the Matminer local order parameter
fingerprint (?) of initialized and relaxed supercells, and the maximum O 2p
band center relative to the Fermi level, which capture their tendency for
phase metastability (?), structural distortion (?), and oxygen
nonstoichiometry (?), respectively. (D to F) ROC–AUC scores of top-performing
2D descriptors in classifying these perovskites as cation-ordered or
disordered before and after filtering them by limiting the range of the
examined properties. $\Delta r_{\mathrm{ion(B)}}$ and $\Delta
n_{\mathrm{ox(B)}}$ are the differences between the ionic radii and oxidation
states of B-site ions, respectively. $S_{\mathrm{conf}}$ and $P_{\mathrm{r}}$
are the configuration entropy (Eq. 2) and the likelihood of generating the
rocksalt configuration (Eq. 1), respectively. The ROC curves of these two 2D
descriptors are provided in Fig. S5.

We verified that these thermodynamic descriptors are physically robust, and
their performance does not obviously degrade as a function of variation in
other features, providing a systematic, comprehensive benchmark between theory
and experiments. The DFT-computed formation energetics of model cation
arrangements for the $190$ A2B’B”O6 perovskites are consistent with the fact
that they have been experimentally synthesized. $81\%$ of the calculated
oxides have energies above the convex hull ($E_{\mathrm{hull}}$) in their
compositional phase diagrams below $0.2\ \mathrm{eV}$ per atom (Fig. 3A). This
range agrees with the previous empirical limit of $E_{\mathrm{hull}}$ for
generating metastable oxides (?). Interestingly, such phase metastability
constrains the performance of atomic descriptors. Filtering out the remaining
$19\%$ perovskites with $E_{\mathrm{hull}}>0.2\ \mathrm{eV}$ boosted the
ROC–AUC of $\\{\Delta r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$ from
$0.81$ to $0.92$ (Fig. 3D). However, no changes in the AUC were shown for
thermodynamic descriptors such as $\\{S_{\mathrm{conf}},P_{\mathrm{r}}\\}$.
This difference can be possibly rationalized by a bias in dataset distribution
when conceiving traditional atomic descriptors, where the focus might have
been disproportionately placed on perovskites that are less metastable and
more synthetically accessible. In contrast, the data-driven evaluation across
a wide oxide space in this work gives rise to more heuristic-free, unbiased
(?) design principles of cation ordering, more independent of the phase
metastability of the composition itself.

Similarly, other structural and compositional features, including structural
distortion and oxygen nonstoichiometry, have been proposed to affect cation
ordering in A2B’B”O6 oxides (?). We quantified the structural distortion in
these oxides using the difference between cubic and DFT-relaxed supercells (?,
?) (Fig. 3B) and estimated their tendency for showing oxygen nonstoichiometry
using the O 2p band center relative to the Fermi level (?) (Fig. 3E). Similar
to $E_{\mathrm{hull}}$, for structural distortion and oxygen nonstoichiometry,
$\\{S_{\mathrm{conf}},P_{\mathrm{r}}\\}$ is also more unbiased than $\\{\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$ (Figs. 3 and S5). Therefore,
we established a co-validation between simulations and experiments and
highlighted the robustness of thermodynamic descriptors for inferring cation
ordering in these multicomponent perovskite oxides.

We further benchmarked the expressivity of multi-dimensional thermodynamic
descriptors and showed that the linear combination of physically interpretable
parameters can well capture the cation ordering in multicomponent oxides.
Specifically, we compared our top-performing descriptors, e.g.,
$\\{S_{\mathrm{conf}},P_{\mathrm{r}},\Delta r_{\mathrm{ion(B)}},\Delta
n_{\mathrm{ox(B)}}\\}$ and $\\{\Delta E_{\mathrm{l,r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$, with those built using the
sure independence screening and sparsifying operator (SISSO) method (?). SISSO
examines up to millions of non-linear mathematical expressions from selected
primary features (Table S1) and algebraic and functional operations, where the
best descriptors can be further discovered (Fig. 4A; Materials and Methods).
Notably, adding non-linear expressivity gives rise to a moderate increase in
descriptor performance for the binary classification of A2B’B”O6 perovskites
into cation-ordered or disordered ones (Fig. 4B). SISSO-learned one-
dimensional (1D) atomic and thermodynamic–atomic descriptors have $86\%$ and
$94\%$ accuracies in ranking cation-ordered perovskites above their disordered
counterparts, which are higher than those of linear models based on
descriptors such as $\\{S_{\mathrm{conf}},P_{\mathrm{r}}\\}$ ($81\%$) and
$\\{S_{\mathrm{conf}},P_{\mathrm{r}},\Delta r_{\mathrm{ion(B)}},\Delta
n_{\mathrm{ox(B)}}\\}$ ($90\%$), respectively. Interestingly, the parameters
in such simple linear descriptors are also the most important primary features
in building the optimal SISSO-derived descriptors (Tables S2 and S3),
demonstrating that these key parameters capture cation ordering well. For
instance, the best 1D thermodynamic–atomic and atomic descriptors are $\Delta
E_{\mathrm{l,r}}\Delta r_{\mathrm{ion(B)}}+\Delta
n_{\mathrm{ox(B)}}\chi_{\mathrm{(A)}}$ and $(\Delta
n_{\mathrm{ox(B)}}+\Delta\chi_{\mathrm{(B)}})\bar{r}_{\mathrm{ion(B)}}r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$,
respectively, which can be combined as a 2D descriptor that accurately
distinguishes cation-ordered perovskites from the disordered ones (Fig. 4C).
In these SISSO-learned descriptors, apart from $\Delta r_{\mathrm{ion(B)}}$
and $\Delta n_{\mathrm{ox(B)}}$, the presence of other atomic parameters,
including the electronegativity of the A-site cations ($\chi_{\mathrm{(A)}}$)
and the average ($\bar{r}_{\mathrm{ion(B)}}$) and product between the ionic
radii of the B-site cations
($r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$), shows
underexplored atomic features worthy of potential studies in future work.
Overall, we systematically evaluated descriptors beyond the linear combination
of atomic and thermodynamic parameters using the SISSO method, validating that
linear descriptors (Figs. 2, S2, and S4) are physically expressive and can
well characterize the cation ordering in multicomponent perovskite oxides.

Fig. 4:  SISSO-learned cation ordering descriptors. (A) Schematic illustration
of the SISSO method, which searches for top-performing descriptors out of up
to millions of candidates using a combinatorial set of primary features and
mathematical operations. SISSO selects the optimal descriptors from these
candidates by minimizing the domain overlap between cation-disordered and
ordered perovskite oxides. (B) Optimal performance of the SISSO method using
logistic regression with five-fold cross-validation, quantified by ROC curves
with AUC in parentheses. Dashed lines show the ROC curves of a random
classifier (AUC $=0.5$). The AUC scores of the SISSO method with different
settings are provided in Table S4. (C) Decision boundary of a 2D descriptor
constructed using the best SISSO-learned 1D atomic and thermodynamic–atomic
descriptors. $P_{\mathrm{ordered}}$ is the predicted likelihood that an oxide
should be classified as cation-ordered through logistic regression. The
definitions of all primary features used for the SISSO analysis are provided
in Table S1.

### Using cation ordering descriptors to boost chemical space exploration

Since the proposed descriptors can elucidate cation ordering, we evaluated how
to use them for boosting the HTVS of multicomponent oxides. The focus was
balancing the computational cost and property prediction accuracy over the
vast configurational space of cation arrangements that will be responsible for
the observed properties at any given composition. Instead of using massive
supercells to characterize the quasirandom cation sublattices of
multicomponent oxides (?, ?), HTVS can be conducted through DFT calculations
of smaller supercells with model cation arrangements (Fig. 1) and quantify
whether one ordered phase will dominate in structures and properties, or
instead, an ensemble of disordered structures will exist, giving a weighted
average of the properties over various possible cation configurations in the
supercell.

We assessed the accuracy and cost of various strategies for selecting limited
representative cation arrangements in A2B’B”O6 perovskites, seeking to
replicate the prediction of weighted properties across all orderings at a
lower cost (Fig. 5). We compared strategies such as selecting a single cation
ordering or multiple tentative orderings. In addition, we evaluated whether
our descriptors can triage oxides expected to be ordered—requiring fewer
property evaluations—from those expected to be disordered, which may require
more exhaustive calculations.

Specifically, we compared mean absolute errors (MAE) of predicting properties,
computed from a weighted average of orderings chosen by various strategies,
with respect to the weighted average of the exhaustive—but prohibitively
expensive—DFT calculations of all configurations (Materials and Methods).
Notably, we focused on two key properties of interest, including the O 2p band
center and bandgap of perovskite oxides, which have been shown to govern their
critical functions, e.g., catalytic activity (?, ?), electrochemical stability
(?, ?), optical absorption (?), intrinsic electrical conductivity (?), and
oxygen surface exchange kinetics (?).

Particularly, we used $\\{\Delta r_{\mathrm{ion(B)}},\Delta
n_{\mathrm{ox(B)}}\\}$ and $\\{\Delta E_{\mathrm{l,r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$ as representative atomic and
thermodynamic–atomic descriptors, respectively, to decide which oxides are
likely disordered and require more calculations. While $S_{\mathrm{conf}}$ and
$P_{\mathrm{r}}$ are more expressive than $\Delta E_{\mathrm{l,r}}$, they
require exhaustive DFT evaluation of all orderings. Therefore, we selected
$\Delta E_{\mathrm{l,r}}$ as the thermodynamic parameter, alongside $\Delta
r_{\mathrm{ion(B)}}$ and $\Delta n_{\mathrm{ox(B)}}$, which are
computationally efficient, as these two parameters can be relatively easily
deduced from perovskite compositions (?).

Fig. 5:  Descriptor-accelerated chemical space exploration by balancing
accuracy and cost. (A and B) Trade-offs between the accuracy and cost of
various descriptor-driven and descriptor-free strategies for sampling the
configurational space of cation arrangements in $190$ A2B’B”O6 perovskite
oxides. The O 2p band center and bandgap are the targeted key properties, with
representative sampling results provided in Figs. S6 and S7, respectively. For
each strategy, its computational cost was quantified based on the number of
required DFT calculations per oxide, and its accuracy was obtained from the
MAE of sampled properties with respect to the computationally expensive ground
truth based on the ensemble average of all configurations. $\Delta
r_{\mathrm{ion(B)}}$ and $\Delta n_{\mathrm{ox(B)}}$ are the differences
between the ionic radii and oxidation states of B-site ions, respectively.
$\Delta E_{\mathrm{l,r}}$ is the energy gap between the layer and rocksalt
arrangements. Error bars represent the standard deviations of $50$ independent
tests of random sampling.

Descriptor-driven selection strategies were found to representatively, yet
efficiently explore the massive configurational space of cation ordering in
multicomponent perovskite oxides. Interestingly, while randomly generating
cation arrangements has been widely leveraged in literature (?, ?), this
strategy can lead to MAEs of $300\ \mathrm{meV}$ (Figs. 5A and S6) and $500\
\mathrm{meV}$ (Figs. 5B and S7) for sampling the O 2p band center and bandgap
of A2B’B”O6 perovskite oxides, respectively, by overweighting the likelihood
of unrealistic orderings. Unfortunately, an error of $300\ \mathrm{meV}$ in
the O 2p band center of oxides can overestimate or underestimate their
electrochemical oxygen evolution or surface exchange rates by one order of
magnitude (?). Similarly, a difference of $500\ \mathrm{meV}$ in the bandgap
can modulate an oxide from being optimal to impractical for solar light
capture and photoelectrochemical water splitting (?). In contrast to random
sampling, manually predefining a cation arrangement for HTVS (?, ?) can give
rise to a much lower error in estimating the cation ordering–averaged
properties. For example, the MAEs can be decreased by one order of magnitude
when focusing on the experimentally dominant configuration of cation-ordered
perovskite oxides, such as the rocksalt arrangement for A2B’B”O6 perovskites
(Fig. 5). If it is computationally affordable to examine more than one
symmetrically inequivalent cation arrangement per oxide, cation ordering
descriptors, e.g., $\\{\Delta E_{\mathrm{l,r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$, can be utilized to triage
ordered perovskite oxides from disordered ones that need further calculations
and decrease the MAEs by another order of magnitude to reach $10\
\mathrm{meV}$ (Fig. 5A) and $9\ \mathrm{meV}$ (Fig. 5B) for sampling the O 2p
band center and bandgap, respectively, with an average cost of three DFT
calculations per oxide. Such descriptor-driven strategies reside at the Pareto
front of accuracy–cost trade-offs, highlighting their crucial roles in
accelerating HTVS by predicting which perovskites should be cation-disordered
and require more extensive DFT exploration of cation configurations than their
ordered counterparts.

## Discussion

In summary, we utilized high-throughput atomistic simulations and machine
learning to build physically interpretable design principles that universally
rationalize and accurately predict the experimental cation ordering in
multicomponent perovskite oxides. Leveraging a dataset of $190$ experimentally
studied A2B’B”O6 double perovskites, we demonstrated that such ordering
descriptors can be obtained from first-principles calculations of model cation
arrangements in computationally inexpensive $2\times 2\times 2$ supercells.
These descriptors can be rationalized as physics-informed parameters such as
thermodynamic likelihood and configurational entropy, outperforming their
traditional atomic counterparts in classifying oxides as cation-ordered or
disordered. Combining such thermodynamic and atomic parameters further leads
to hybrid descriptors that can correctly rank up to $93\%$ perovskite
compositions between cation-ordered and disordered, offering an experiment-
backed, systematic validation of atomistic simulations and data science. We
verified that these design principles are physically robust and expressive and
can be applied to other multicomponent perovskite oxides with more exotic
compositions and orderings. Lastly, we highlighted that such physics-driven
descriptors can be leveraged to boost the high-throughput exploration of
multicomponent oxides across a vast compositional space by facilitating an
optimal balance between the breadth and cost of computationally expensive DFT
sampling for different cation arrangements in these complex oxides.

The design principles of cation ordering established in this work can be
potentially extended for rationalizing and inferring the chemical and
structural orders in other multicomponent oxide chemistries. High-entropy
oxides represent an exciting, yet largely uncharted chemical space, which
creates exotic physical and chemical properties and flips traditional research
paradigms by introducing profound configurational disorder in these new
materials (?). While these high-entropy oxides generally do not have long-
range ordering, they can still exhibit local clustering (?), short-range
ordering (?), and site selectivity (?). Unfortunately, most previous studies
on high-entropy oxides have so far overlooked such local chemical and
structural orders and the unique role of these local orders in influencing key
material properties (?). This limitation arises from the difficulty of
quantitatively characterizing such chemical and structural orders, as their
spatial scales evade most available experimental techniques (?). To conquer
this task, it will be advantageous to leverage simulations, which, in theory,
have full access to determining a realistic atom-by-atom picture of these
high-entropy oxides but, in practice, face the challenge of accurately and
efficiently sampling their high-dimensional configurational spaces. This work
can serve as a blueprint for future studies to elucidate the correlated
arrangements of cations in high-entropy oxides and predict their degree of
order or disorder at the atomistic level. From a broad perspective, we aim to
couple physics-informed principles with data-driven tools in future work to
simultaneously navigate the compositional and configurational spaces of high-
entropy materials and develop effective design principles to accelerate their
global optimization.

## Materials and Methods

### Dataset curation

A dataset of $190$ experimentally examined A2B’B”O6 compositions was curated
according to a recent review (?). We focused on oxides that are synthesizable
under ambient pressure, where all but one known cation-ordered A2B’B”O6
perovskites have B-site sublattices with rocksalt ordering (?, ?). This only
composition with B-site layered ordering under ambient pressure was not
included for preventing too-massive data imbalance in training and assessing
machine learning models. Overall, this dataset has $119$ cation-ordered oxides
with rocksalt ordering and $71$ cation-disordered oxides. Among these $119$
perovskite oxides, eight of them only have partially ordered cation
arrangements, as shown by their low ($<0.5$) experimentally measured long-
range order parameters (?). Apart from this dataset, an additional evaluation
set with less common compositions and orderings was built according to
experimental literature (?, ?, ?, ?, ?).

### DFT calculations

Periodic plane-wave spin-polarized DFT calculations were performed to assess
the electronic structures and energetics of multicomponent perovskite oxides
and their cation ordering. We used Perdew–Burke–Ernzerhof (PBE) functional (?)
as implemented in the Vienna Ab initio Simulation Package (?, ?) and projector
augmented wave method (?) for the description of core-electron interaction,
and the plane-wave cutoff was set to $520\ \mathrm{eV}$. DFT calculations were
conducted with Hubbard $U$ correction for V 3d ($U_{\mathrm{V}}=3.25\
\mathrm{eV}$), Cr 3d ($U_{\mathrm{Cr}}=3.7\ \mathrm{eV}$), Mn 3d
($U_{\mathrm{Mn}}=3.9\ \mathrm{eV}$), Fe 3d ($U_{\mathrm{Fe}}=5.3\
\mathrm{eV}$), Co 3d ($U_{\mathrm{Co}}=3.32\ \mathrm{eV}$), Ni 3d
($U_{\mathrm{Ni}}=6.2\ \mathrm{eV}$), Mo 4d ($U_{\mathrm{Mo}}=4.38\
\mathrm{eV}$), and W 5d ($U_{\mathrm{W}}=6.2\ \mathrm{eV}$) electrons, where
the $U$ values were optimized by fitting the experimental formation enthalpies
of binary oxides (?). DFT calculations were initialized with high-spin
ferromagnetic states in order to use a consistent and tractable set of
magnetic structures. These computational settings are fully compatible with
the Materials Project database (?), as the Materials Project represents a good
standard for high-throughput materials discoveries, and we can utilize this
expansive database to derive specific properties of multicomponent perovskite
oxides (e.g., $E_{\mathrm{hull}}$). Utilizing more advanced density functional
and more diverse initial magnetic configurations might improve the accuracy of
DFT calculations but are too computationally expensive for high-throughput
exploration. Therefore, in this work, we focused on PBE-level calculations
with high-spin ferromagnetic initialization and showed that they could well
capture experimental ordering across a wide oxide space.

We used an in-house automated high-throughput DFT pipeline for structure
optimization and electronic structure calculation. Initial $2\times 2\times 2$
cubic supercells were built by generating all symmetrically distinct cation
arrangements with the Atomic Simulation Environment (?) and Python Materials
Genomics (pymatgen) (?). The initialized lattice parameters were set to be $8\
\mathrm{\mathring{A}}$, and all atoms in these initialized cubic supercells
could be displaced by a distance randomly sampled from a uniform distribution
with a range between $0.01\ \mathrm{\mathring{A}}$ and $0.1\
\mathrm{\mathring{A}}$ to break symmetry and boost convergence. The
convergence threshold for electronic steps was $10^{-6}\ \mathrm{eV}$ per unit
cell, and the residual forces on all atoms were lower than $10^{-2}\
\mathrm{eV\ \mathring{A}}$. O 2p band centers were determined by taking the
centroid of the O 2p projected density of states relative to the Fermi level.
$E_{\mathrm{hull}}$ was computed using the pymatgen (?) by combining the DFT
energies of multicomponent oxides with all data entries within the examined
compositional space from the Materials Project (?). The degree of structural
distortion before and after DFT relaxation was estimated by the cosine
distances between the Matminer local order parameter fingerprint (?) of
initialized and relaxed supercells based on a method developed in a recent
work (?).

### Descriptor construction

Cation ordering descriptors were constructed using material parameters, i.e.,
$\Delta E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$, $P_{\mathrm{r}}$,
$\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and $\Delta
n_{\mathrm{ox(B)}}$, where these parameters were used as individual 1D
descriptors and combined as multi-dimensional descriptors. First,
$P_{\mathrm{r}}$ and $S_{\mathrm{conf}}$ were derived from the DFT-computed
energies of all symmetrically distinct cation arrangements based on Eqs. 1 and
2, respectively. We set $T=1300\ \mathrm{K}$ based on an extensive survey on
the synthesis conditions of perovskite oxides (?), where long-term calcination
at comparable temperatures followed by rapid quenching has been widely used in
experimental literature to kinetically trap these oxides at their high-
temperature, thermodynamically favorable perovskite phases (?, ?). Although
based on convention (?), the configurational entropy of cation arrangements
should be:

$\hat{S}_{\mathrm{conf}}=-\sum_{i=1}^{n}\Omega_{i}\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}\ln{\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}}$
(3)

the range of $\hat{S}_{\mathrm{conf}}$ depends too strongly on the degeneracy.
Specifically, if the energies of all cation arrangements are equal,
$\hat{S}_{\mathrm{conf}}$ is maximized as $\ln{N_{\mathrm{total}}}$, where
$N_{\mathrm{total}}$ is the total number of all possible cation configurations
($N_{\mathrm{total}}=70$ for A2B’B”O6 in Fig. 1):

$N_{\mathrm{total}}=\sum_{k=1}^{N_{\mathrm{unique}}}\Omega_{k}$ (4)

On the other hand, if a cation arrangement $\alpha$ is much more
thermodynamically favorable than its symmetrically inequivalent counterparts,
$\hat{S}_{\mathrm{conf}}$ is minimized as $\ln{\Omega_{\alpha}}$. To
facilitate the intuitive comparison of the configurational entropy, we further
normalized it to make it dimensionless:

$\begin{gathered}S_{\mathrm{conf}}=\frac{\hat{S}_{\mathrm{conf}}}{\ln{N_{\mathrm{total}}}}=\frac{\hat{S}_{\mathrm{conf}}}{\ln{\sum_{k=1}^{N_{\mathrm{unique}}}\Omega_{k}}}\\\
=-\sum_{i=1}^{n}\frac{\Omega_{i}}{\ln{\sum_{k=1}^{N_{\mathrm{unique}}}\Omega_{k}}}\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}\ln{\frac{\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}}\end{gathered}$
(5)

with a range between $S_{\mathrm{residual}}$ and $1$, where
$S_{\mathrm{residual}}$ is the residual entropy (?) originating from the
configurational multiplicity of the symmetrically distinct ground-state
arrangement $\alpha$:

$S_{\mathrm{residual}}=\frac{\ln{\Omega_{\alpha}}}{\ln{N_{\mathrm{total}}}}=\frac{\ln{\Omega_{\alpha}}}{\ln{\sum_{k=1}^{N_{\mathrm{unique}}}\Omega_{k}}}$
(6)

Second, $\Delta E_{\mathrm{l,r}}$ was calculated based on the formation
energies of the layer ($E_{\mathrm{layered}}$) and rocksalt cation
arrangements ($E_{\mathrm{rocksalt}}$):

$\Delta
E_{\mathrm{l,r}}=\frac{E_{\mathrm{layered}}-E_{\mathrm{rocksalt}}}{k_{\mathrm{B}}T}$
(7)

where the energy is normalized by $k_{\mathrm{B}}T$ to make it dimensionless.
Lastly, $\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and $\Delta
n_{\mathrm{ox(B)}}$ were estimated by calculating the differences between the
Pauling electronegativities (?), Shannon effective ionic radii (?), and
nominal oxidation states of the two different B-site cations in double
perovskites, respectively. Similar to $\Delta E_{\mathrm{l,r}}$, $\Delta
r_{\mathrm{ion(B)}}$ is normalized by the ionic radius of an oxygen anion
($r_{\mathrm{ion(O^{2-})}}=1.40\ \mathrm{\mathring{A}}$) to make it
dimensionless.

### Logistic regression

A logistic regression model was utilized to assess the performance of
descriptors in classifying perovskite oxides into cation-ordered and
disordered structures. Material parameters, including $\Delta
E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$, $P_{\mathrm{r}}$,
$\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and $\Delta
n_{\mathrm{ox(B)}}$, were used to construct one- or multi-dimensional
descriptors as input features. For an input feature $(x_{1},\ldots,x_{M})$,
where $M$ is its dimension, the model predicts the likelihood that an oxide
should be classified as cation-ordered:

$P_{\mathrm{ordered}}=\frac{1}{1+\exp(-\sum_{i=1}^{M}\theta_{i}x_{i}-\theta_{0})}$
(8)

where $\theta_{0},\ldots,\theta_{M}$ are learnable parameters. An oxide is
predicted as cation-ordered if $P_{\mathrm{ordered}}>0.5$ and disordered if
$P_{\mathrm{ordered}}<0.5$. The logistic regression model was trained by
minimizing its loss function in the form of negative log-likelihood (NLL) with
$L_{2}$ regularization:

$l_{\mathrm{NLL}}=\lambda\sum_{i=0}^{M}\theta_{i}^{2}-\sum_{j=1}^{N_{\mathrm{oxide}}}(y_{\mathrm{ordered},j}\ln{P_{\mathrm{ordered},j}}+(1-y_{\mathrm{ordered},j})\ln(1-P_{\mathrm{ordered},j}))$
(9)

where $\lambda$ is the regularization hyperparameter, $N_{\mathrm{oxide}}$ is
the number of oxides, $y_{\mathrm{ordered},j}$ is the label indicating whether
an oxide $j$ is experimentally cation-ordered ($y_{\mathrm{ordered},j}=1$ if
ordered and $y_{\mathrm{ordered},j}=0$ if disordered), and
$P_{\mathrm{ordered},j}$ is the predicted likelihood if such an oxide should
be classified as cation-ordered through logistic regression (Eq. 8).

The model was trained using the scikit-learn (?) and evaluated by both ROC–AUC
scores and confusion matrices. Five-fold stratified cross-validation was
utilized to rigorously quantify the accuracy and generalizability of model
performance for predicting the cation ordering of unseen multicomponent
perovskite oxides, given the random splits and the unbalanced numbers of
experimentally cation-ordered and disordered oxides. The logistic regression
model was first trained by combining four splits ($80\%$) and then tested on
the one hold-out split ($20\%$). All input features were normalized by the
mean and standard deviation of training sets. This process was iterated
throughout the five folds, where only the $P_{\mathrm{ordered}}$ of test sets
was collected for further assessment. ROC curves and AUC scores were obtained
by averaging over the five folds, while the confusion matrices were based on
the combined test set performance of these five folds. The ROC curves were
obtained from the true positive rate (TPR) and false positive rate (FPR) for
classifying perovskites as cation-ordered and disordered at various decision
thresholds. The TPR is the percentage of experimentally cation-ordered oxides
that are correctly predicted as ordered, while the FPR is the percentage of
experimentally cation-disordered oxides that are incorrectly predicted as
ordered. A perfect classifier maximizes the TPR as $1$ and minimizes the FPR
as $0$, leading to an AUC of $1$. In contrast, a trivial classifier that
infers cation ordering by randomly sampling a Bernoulli distribution based on
the ratio of cation-ordered and disordered perovskites should always have
equal TPR and FPR, resulting in an AUC of $0.5$.

### SISSO

We additionally leveraged the SISSO method (?) to assess the space of
descriptors beyond the linear combination of atomic and thermodynamic
parameters. Using an iterative approach, SISSO constructs non-linear
expressions based on selected primary features and mathematical operations.
These features were hypothesized to be potentially relevant for describing the
cation ordering in multicomponent perovskite oxides and thus used as a
starting point to build complex descriptors. The examined primary features are
$\Delta E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$, $P_{\mathrm{r}}$,
$\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and $\Delta
n_{\mathrm{ox(B)}}$, as well as nine additional atomic parameters, including
the electronegativity, ionic radius, and oxidation state of the A-site cations
and the average and product between the electronegativities, ionic radii, and
oxidation states of the B-site cations (Table S1). These combinations of
B-site atomic parameters were used such that the constructed descriptors are
independent of the order of B’ and B” cations in A2B’B”O6 perovskites. All
such primary features are dimensionless, and thus they can be combined with a
set of selected algebraic and functional operations recursively to expand the
descriptor space beyond linearity. The assessed mathematical operations
include $+$, $-$, $\times$, $\div$, $|\ |$, -1, 2, 3, 4, 1/2 1/3, 1/4, $\exp$,
$\log$, $\sin$, and $\cos$. In total, we conducted two rounds of operations,
leading to $498420$ candidates from only atomic features and $996440$
candidates from atomic and thermodynamic features after excluding numerically
unstable ones (e.g., infinity).

Utilizing these features and operations, SISSO downselected the optimal
descriptors from up to millions of candidates (Fig. 4A). We used sure
independence screening (?) to identify $1000$ top-performing non-linear
expressions that minimize the domain overlap between cation-disordered and
ordered perovskite oxides. Then, we utilized the sparsifying operator approach
to further select the optimal SISSO-derived descriptors, where we experimented
with two types of previous implementations based on convex hull (?) or a
decision tree with a depth of $2$ (?), respectively. We assessed both
implementations with and without using balanced class weights. Thus, when
accounting for all settings, we carried out, in total, $16$ SISSO experiments
(Table S4). As all such SISSO experiments were conducted with five-fold cross-
validation, we focused on the AUC of the hold-out test splits in order to
evaluate the generalizability of SISSO-learned descriptors for unseen
perovskites. Overall, we found that using balanced class weights gave rise to
better performance. Moreover, the convex hull method worked better for
building atomic descriptors, while the decision tree method performed better
in finding thermodynamic–atomic descriptors. We summarized the best SISSO
performance of Table S4 in Fig. 4.

We found $\Delta E_{\mathrm{l,r}}\Delta r_{\mathrm{ion(B)}}+\Delta
n_{\mathrm{ox(B)}}\chi_{\mathrm{(A)}}$ and $(\Delta
n_{\mathrm{ox(B)}}+\Delta\chi_{\mathrm{(B)}})\bar{r}_{\mathrm{ion(B)}}r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$
to be the best 1D SISSO-learned descriptors in four out of five folds of
cross-validation when selecting the optimal thermodynamic–atomic and atomic
descriptors, respectively. This result shows that the downselection of such
two 1D descriptors is relatively independent of the choices of training and
test sets in cross-validation. We also examined the performance of the SISSO
method with increased dimensionality using the residuals (i.e., those
misclassified by the best 1D descriptors) in the search for a second dimension
(?). Unfortunately, this second dimension was found to lead to negligible
improvement in classification performance compared with the corresponding 1D
descriptor (Table S4). Thus, we only focused on the best 1D descriptors in
Fig. 4.

Lastly, we note that, while we used SISSO to explore a space of up to millions
of descriptor candidates, the potential for overfitting is minimal, as for 1D
or 2D SISSO-derived descriptors, the SISSO method has only one or two
independent learnable parameters that define the decision boundary,
respectively. Moreover, finding different descriptors in five-fold cross-
validation does not imply overfitting, as the best expressions from SISSO
experiments reflect the intrinsically approximate nature of these expressions
and also the unavoidable correlations between various descriptor candidates.
In addition, another possible source of overfitting can come from a too-large
dimensionality of SISSO-learned descriptors. However, this risk of overfitting
is minimal in this work since we only examined 1D and 2D descriptors using the
SISSO method. A more detailed discussion on these points was well presented in
a previous study (?).

### Chemical space exploration

We tested various descriptor-driven and descriptor-free strategies to sample
the configurational space of cation arrangements in $190$ A2B’B”O6 perovskites
and quantify the accuracy and cost of such strategies for the HTVS of
multicomponent oxides. These strategies are composed of a set of potential
actions, including sampling a specific symmetrically unique cation
configuration deterministically, selecting a configuration randomly from a
pool of cation arrangements, and leveraging descriptors to infer cation
ordering and decide sampling methods accordingly. The computational cost was
quantified based on the number of required DFT calculations per oxide. The
accuracy was obtained from the MAE of the sampled properties
($\Phi_{\mathrm{sampled}}$) with respect to the computational ground truth
based on the ensemble average of all possible configurations:

$\Phi_{\mathrm{true}}=\sum_{i=1}^{N_{\mathrm{unique}}}\Phi_{i}P_{i}=\frac{\sum_{i=1}^{N_{\mathrm{unique}}}\Phi_{i}\Omega_{i}\exp{\left(-\frac{E_{i}}{k_{\mathrm{B}}T}\right)}}{\sum_{j=1}^{N_{\mathrm{unique}}}\Omega_{j}\exp{\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}}$
(10)

where $\Phi_{i}$ is the property (e.g., the O 2p band center or bandgap) of a
symmetrically distinct configuration $i$. When selecting a configuration
randomly from a number of candidates, $\Phi_{\mathrm{sampled}}$ was computed
by assuming all other unsampled cation arrangements have the same $E_{i}$ and
$\Phi_{i}$ as this sampled arrangement. When sampling both the rocksalt and
layered configurations, all unsampled arrangements were assumed to have the
same $E_{i}$ and $\Phi_{i}$ as the layered arrangement. For descriptor-driven
strategies, five-fold stratified cross-validation was leveraged to examine the
generalizability of descriptors for unseen perovskites by training the model
on four splits and utilizing the prediction for the hold-out split. For
strategies involving random sampling, $50$ independent tests were conducted to
quantify their statistically averaged performance.

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Acknowledgments

Funding: This project was funded by the Advanced Research Projects
Agency–Energy (ARPA-E), U.S. Department of Energy, under award number DE-
AR0001220. This work used Expanse at San Diego Supercomputer Center through
allocation DMR200068 from the Advanced Cyberinfrastructure Coordination
Ecosystem: Services & Support (ACCESS) program, which is supported by National
Science Foundation grants 2138259, 2138286, 2138307, 2137603, and 2138296.
This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility located at Lawrence Berkeley National Laboratory, operated under
Contract DE-AC02-05CH11231 using NERSC award m4074. We acknowledge the MIT
Engaging cluster at the Massachusetts Green High-Performance Computing Center
(MGHPCC) for providing high-performance computing resources. Author
contributions: J.P. and R.G.-B. conceived the original idea. J.P. designed the
studies. J.P. and J.D. performed the studies and analyzed the results. J.P.
drafted the manuscript. All authors edited the manuscript. R.G.-B. supervised
the project. Competing interests: The authors declare no competing interests.
Data and materials availability: A repository for reproducing this work is
available at https://github.com/learningmatter-mit/Perovskite-Ordering-
Descriptors.

List of supplementary materials

Figs. S1 to S7
Tables S1 to S4

Supplementary Materials for

Data-Driven, Physics-Informed Descriptors of Cation Ordering in Multicomponent
Oxides

Jiayu Peng, James Damewood, Rafael Gómez-Bombarelli∗

∗Corresponding author. Email<EMAIL_ADDRESS>(R.G.-B.)

This PDF file includes:

Figs. S1 to S7

Tables S1 to S4

Fig. S1:  Dataset statistics. Heat map of the occurrences of elements in
A-site and B-site cations for the experimental dataset of $190$ A2B’B”O6
double perovskites. These cations include a wide variety of elements that
cover (but are not limited to) commonly seen A-site (alkaline earth and rare
earth metals) and B-site ions (transition metals) in perovskite oxides,
highlighting that this dataset contains oxide compositions across a broad
space of multicomponent perovskites.  Fig. S2:  Additional performance. (A to
J) Confusion matrices for classifying $190$ A2B’B”O6 perovskites as cation-
ordered and disordered using logistic regression with five-fold cross-
validation. $\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and
$\Delta n_{\mathrm{ox(B)}}$ are the differences between the
electronegativities, ionic radii, and oxidation states of B-site ions,
respectively. $\Delta E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$, and
$P_{\mathrm{r}}$ are the energy gap between the layer and rocksalt
arrangements, the configuration entropy, and the likelihood of generating the
rocksalt configuration, respectively. (K) Decision boundary of the three-
dimensional descriptor $\\{\Delta E_{\mathrm{l,r}},\Delta
r_{\mathrm{ion(B)}},\Delta n_{\mathrm{ox(B)}}\\}$. $P_{\mathrm{ordered}}$ is
the predicted probability that an oxide should be classified as cation-ordered
through logistic regression.  Fig. S3:  Correlation matrix. Pearson
correlation coefficients between different parameters for the experimental
dataset of $190$ A2B’B”O6 double perovskites. $\Delta\chi_{\mathrm{(B)}}$,
$\Delta r_{\mathrm{ion(B)}}$, and $\Delta n_{\mathrm{ox(B)}}$ are the
differences between the electronegativities, ionic radii, and oxidation states
of B-site ions, respectively. $\Delta E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$,
and $P_{\mathrm{r}}$ are the energy gap between the layer and rocksalt
arrangements, the configuration entropy, and the likelihood of generating the
rocksalt configuration, respectively.  Fig. S4:  Descriptor performance
without experimentally partially cation-ordered oxides. (A to C) $182$
A2B’B”O6 perovskites were left after filtering out eight experimentally
partially cation-ordered oxides. Performance of multi-dimensional descriptors
in classifying these $182$ oxides as cation-ordered and disordered using
logistic regression with five-fold cross-validation, quantified by receiver
operating characteristic (ROC) curves with area under the curve (AUC) in
parentheses. Dashed lines show the ROC curves of a random classifier (AUC
$=0.5$). $\Delta\chi_{\mathrm{(B)}}$, $\Delta r_{\mathrm{ion(B)}}$, and
$\Delta n_{\mathrm{ox(B)}}$ are the differences between the
electronegativities, ionic radii, and oxidation states of B-site ions,
respectively. $\Delta E_{\mathrm{l,r}}$, $S_{\mathrm{conf}}$, and
$P_{\mathrm{r}}$ are the energy gap between the layer and rocksalt
arrangements, the configuration entropy, and the likelihood of generating the
rocksalt configuration, respectively. (D to F) Confusion matrices of top-
performing multi-dimensional descriptors after filtering out eight
experimentally partially cation-ordered oxides.  Fig. S5:  Additional
sensitivity analysis. ROC curves of top-performing two-dimensional descriptors
in classifying $190$ A2B’B”O6 perovskite oxides as cation-ordered or
disordered before and after filtering them by limiting the range of selected
density functional theory (DFT)–computed properties, including the minimum
energy above the convex hull, the maximum cosine distance between the Matminer
local order parameter fingerprint (?) of initialized and relaxed supercells,
and the maximum O 2p band center relative to the Fermi level, which capture
their tendency for phase metastability (?), structural distortion (?), and
oxygen nonstoichiometry (?), respectively. $\Delta r_{\mathrm{ion(B)}}$ and
$\Delta n_{\mathrm{ox(B)}}$ are the differences between the ionic radii and
oxidation states of B-site ions, respectively. $S_{\mathrm{conf}}$ and
$P_{\mathrm{r}}$ are the configuration entropy and the likelihood of
generating the rocksalt configuration, respectively.  Fig. S6:  Descriptor-
driven and descriptor-free sampling of O 2p band center. Trade-offs between
the accuracy and cost of different strategies for sampling the configurational
space of cation arrangements in $190$ A2B’B”O6 oxides. For each strategy, its
computational cost was quantified based on the number of required DFT
calculations per oxide, and its accuracy was obtained from the mean absolute
error (MAE) of sampled properties with respect to the computationally
expensive ground truth based on the ensemble average of all configurations.
$\Delta r_{\mathrm{ion(B)}}$ and $\Delta n_{\mathrm{ox(B)}}$ are the
differences between the ionic radii and oxidation states of B-site ions,
respectively. $\Delta E_{\mathrm{l,r}}$ is the energy gap between the layer
and rocksalt arrangements.  Fig. S7:  Descriptor-driven and descriptor-free
sampling of bandgap. Trade-offs between the accuracy and cost of different
strategies for sampling the configurational space of cation arrangements in
$190$ A2B’B”O6 oxides. For each strategy, its computational cost was
quantified based on the number of required DFT calculations per oxide, and its
accuracy was obtained from the MAE of sampled properties with respect to the
computationally expensive ground truth based on the ensemble average of all
configurations. $\Delta r_{\mathrm{ion(B)}}$ and $\Delta n_{\mathrm{ox(B)}}$
are the differences between the ionic radii and oxidation states of B-site
ions, respectively. $\Delta E_{\mathrm{l,r}}$ is the energy gap between the
layer and rocksalt arrangements.

Table S1:  Atomic and thermodynamic primary features utilized for building
descriptors using the sure independence screening and sparsifying operator
(SISSO) method.

Feature | Definition
---|---
$\chi_{\mathrm{(A)}}$ | Pauling electronegativity of the A-site cations
$\Delta\chi_{\mathrm{(B)}}$ | Difference between the Pauling electronegativities of the B-site cations
$\bar{\chi}_{\mathrm{(B)}}$ | Average between the Pauling electronegativities of the B-site cations
$\chi_{\mathrm{(B^{\prime})}}\chi_{\mathrm{(B^{\prime\prime})}}$ | Product between the Pauling electronegativities of the B-site cations
$r_{\mathrm{ion(A)}}$ | Shannon effective ionic radius of the A-site cations1
$\Delta r_{\mathrm{ion(B)}}$ | Difference between the Shannon effective ionic radii of the B-site cations1
$\bar{r}_{\mathrm{ion(B)}}$ | Average between the Shannon effective ionic radii of the B-site cations1
$r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$ | Product between the Shannon effective ionic radii of the B-site cations1
$n_{\mathrm{ox(A)}}$ | Nominal oxidation state of the A-site cations
$\Delta n_{\mathrm{ox(B)}}$ | Difference between the nominal oxidation states of the B-site cations
$\bar{n}_{\mathrm{ox(B)}}$ | Average between the nominal oxidation states of the B-site cations
$n_{\mathrm{ox(B^{\prime})}}n_{\mathrm{ox(B^{\prime\prime})}}$ | Product between the nominal oxidation states of the B-site cations
$\Delta E_{\mathrm{l,r}}$ | Energy gap between the layer and rocksalt arrangements2
$S_{\mathrm{conf}}$ | Configurational entropy of all possible cation arrangements3
$P_{\mathrm{r}}$ | Thermodynamic likelihood of forming the rocksalt arrangement

  * 1

Normalized by the ionic radius of an oxygen anion
($r_{\mathrm{ion(O^{2-})}}=1.40\ \mathrm{\mathring{A}}$) to make it
dimensionless.

  * 2

Normalized by $k_{\mathrm{B}}T$ to make it dimensionless, where
$k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is the temperature
($T=1300\ \mathrm{K}$).

  * 3

Normalized by $\ln{N_{\mathrm{total}}}$ to make it dimensionless, where
$N_{\mathrm{total}}$ is the total number of all possible cation
configurations.

Table S2:  Importance of different features in constructing the best SISSO-
learned one-dimensional atomic descriptor.

Type | Feature | Relative importance1
---|---|---
All $12$ primary features | $\Delta n_{\mathrm{ox(B)}}$ | $0.261$
$\bar{r}_{\mathrm{ion(B)}}$ | $0.114$
$\Delta\chi_{\mathrm{(B)}}$ | $0.105$
$r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$ | $0.081$
$\bar{n}_{\mathrm{ox(B)}}$ | $0.081$
$n_{\mathrm{ox(A)}}$ | $0.064$
$\Delta r_{\mathrm{ion(B)}}$ | $0.047$
$n_{\mathrm{ox(B^{\prime})}}n_{\mathrm{ox(B^{\prime\prime})}}$ | $0.040$
$\bar{\chi}_{\mathrm{(B)}}$ | $0.037$
$\chi_{\mathrm{(B^{\prime})}}\chi_{\mathrm{(B^{\prime\prime})}}$ | $0.036$
$\chi_{\mathrm{(A)}}$ | $0.018$
$r_{\mathrm{ion(A)}}$ | $0.012$
Top $12$ secondary features | $\Delta n_{\mathrm{ox(B)}}+\Delta\chi_{\mathrm{(B)}}$ | $0.083$
$\Delta n_{\mathrm{ox(B)}}+\bar{r}_{\mathrm{ion(B)}}$ | $0.050$
$\Delta n_{\mathrm{ox(B)}}+\bar{n}_{\mathrm{ox(B)}}$ | $0.031$
$\bar{n}_{\mathrm{ox(B)}}+\Delta n_{\mathrm{ox(B)}}$ | $0.030$
$\Delta n_{\mathrm{ox(B)}}+r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$ | $0.030$
$\Delta n_{\mathrm{ox(B)}}+\Delta r_{\mathrm{ion(B)}}$ | $0.025$
$\Delta n_{\mathrm{ox(B)}}-n_{\mathrm{ox(A)}}$ | $0.021$
$n_{\mathrm{ox(A)}}-\Delta n_{\mathrm{ox(B)}}$ | $0.020$
$\exp(\Delta n_{\mathrm{ox(B)}})$ | $0.015$
$\frac{\Delta n_{\mathrm{ox(B)}}}{n_{\mathrm{ox(A)}}}$ | $0.014$
$\Delta n_{\mathrm{ox(B)}}\bar{n}_{\mathrm{ox(B)}}$ | $0.013$
$\bar{n}_{\mathrm{ox(B)}}\Delta n_{\mathrm{ox(B)}}$ | $0.013$

  * 1

Based on the number of occurrences in the $1000$ top-performing non-linear
expressions minimizing the domain overlap between cation-disordered and
ordered oxides identified by sure independence screening.

Table S3:  Importance of different features in constructing the best SISSO-
learned one-dimensional thermodynamic–atomic descriptor.

Type | Feature | Relative importance1
---|---|---
All $15$ primary features | $\Delta n_{\mathrm{ox(B)}}$ | $0.278$
$S_{\mathrm{conf}}$ | $0.142$
$P_{\mathrm{r}}$ | $0.092$
$\bar{r}_{\mathrm{ion(B)}}$ | $0.070$
$r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$ | $0.052$
$\Delta\chi_{\mathrm{(B)}}$ | $0.038$
$\Delta r_{\mathrm{ion(B)}}$ | $0.037$
$\bar{n}_{\mathrm{ox(B)}}$ | $0.035$
$n_{\mathrm{ox(A)}}$ | $0.027$
$\bar{\chi}_{\mathrm{(B)}}$ | $0.017$
$\chi_{\mathrm{(B^{\prime})}}\chi_{\mathrm{(B^{\prime\prime})}}$ | $0.016$
$n_{\mathrm{ox(B^{\prime})}}n_{\mathrm{ox(B^{\prime\prime})}}$ | $0.015$
$\Delta E_{\mathrm{l,r}}$ | $0.014$
$\chi_{\mathrm{(A)}}$ | $0.009$
$r_{\mathrm{ion(A)}}$ | $0.007$
Top $15$ secondary features | $\Delta n_{\mathrm{ox(B)}}-S_{\mathrm{conf}}$ | $0.061$
$\Delta n_{\mathrm{ox(B)}}+P_{\mathrm{r}}$ | $0.059$
$\Delta n_{\mathrm{ox(B)}}+\Delta\chi_{\mathrm{(B)}}$ | $0.035$
$\Delta n_{\mathrm{ox(B)}}+\bar{r}_{\mathrm{ion(B)}}$ | $0.034$
$\exp(\Delta n_{\mathrm{ox(B)}})$ | $0.027$
$\Delta n_{\mathrm{ox(B)}}+r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}$ | $0.023$
$\Delta r_{\mathrm{ion(B)}}\Delta E_{\mathrm{l,r}}$ | $0.020$
$\Delta n_{\mathrm{ox(B)}}^{4}$ | $0.019$
$\Delta n_{\mathrm{ox(B)}}^{3}$ | $0.019$
$\Delta n_{\mathrm{ox(B)}}^{2}$ | $0.018$
$\Delta n_{\mathrm{ox(B)}}\Delta n_{\mathrm{ox(B)}}$ | $0.018$
$\Delta n_{\mathrm{ox(B)}}+\Delta r_{\mathrm{ion(B)}}$ | $0.017$
$\frac{\bar{r}_{\mathrm{ion(B)}}}{S_{\mathrm{conf}}}$ | $0.016$
$\frac{r_{\mathrm{ion(B^{\prime})}}r_{\mathrm{ion(B^{\prime\prime})}}}{S_{\mathrm{conf}}}$ | $0.016$
$\Delta n_{\mathrm{ox(B)}}+\Delta n_{\mathrm{ox(B)}}$ | $0.014$

  * 1

Based on the number of occurrences in the $1000$ top-performing non-linear
expressions minimizing the domain overlap between cation-disordered and
ordered oxides identified by sure independence screening.

Table S4:  Performance of the SISSO method with different settings.

Type | Dimension | Method | ROC–AUC
---|---|---|---
Atomic | $1$ | Convex hull, non-weighted | $0.8635$
Convex hull, weighted | $0.8644$
Decision tree, non-weighted | $0.8068$
Decision tree, weighted | $0.8172$
$2$ | Convex hull, non-weighted | $0.8117$
Convex hull, weighted | $0.8291$
Decision tree, non-weighted | $0.8044$
Decision tree, weighted | $0.8032$
Thermodynamic–atomic | $1$ | Convex hull, non-weighted | $0.9211$
Convex hull, weighted | $0.9432$
Decision tree, non-weighted | $0.9424$
Decision tree, weighted | $0.9435$
$2$ | Convex hull, non-weighted | $0.9115$
Convex hull, weighted | $0.9307$
Decision tree, non-weighted | $0.9197$
Decision tree, weighted | $0.9442$