111J.Y. and P.M. contributed equally to this work

# Discovering Novel Halide Perovskite Alloys using Multi-Fidelity Machine
Learning and Genetic Algorithm

Jiaqi Yang  Materials Engineering, Purdue University, 701 W Stadium Ave, West
Lafayette, 47907, Indiana, USA  Panayotis Manganaris  Materials Engineering,
Purdue University, 701 W Stadium Ave, West Lafayette, 47907, Indiana, USA
Arun Mannodi-Kanakkithodi  Materials Engineering, Purdue University, 701 W
Stadium Ave, West Lafayette, 47907, Indiana, USA<EMAIL_ADDRESS>
###### Abstract

Expanding the pool of stable halide perovskites with attractive optoelectronic
properties is crucial to addressing current limitations in their performance
as photovoltaic (PV) absorbers. In this article, we demonstrate how a high-
throughput density functional theory (DFT) dataset of halide perovskite alloys
can be used to train accurate surrogate models for property prediction and
subsequently perform inverse design using genetic algorithm (GA). Our dataset
consists of decomposition energies, band gaps, and photovoltaic efficiencies
of nearly 800 pure and mixed composition ABX3 compounds from both the GGA-PBE
and HSE06 functionals, and are combined with $\sim$ 100 experimental data
points collected from the literature. Multi-fidelity random forest regression
models are trained on the DFT + experimental dataset for each property using
descriptors that one-hot encode composition, phase, and fidelity, and
additionally include well-known elemental or molecular properties of species
at the A, B, and X sites. Rigorously optimized models are deployed for
experiment-level prediction over > 150,000 hypothetical compounds, leading to
thousands of promising materials with low decomposition energy, band gap
between 1 and 2 eV, and efficiency > 15%. Surrogate models are further
combined with GA using an objective function to maintain chemical feasibility,
minimize decomposition energy, maximize PV efficiency, and keep band gap
between 1 and 2 eV; hundreds more optimal compositions and phases are thus
discovered. We present an analysis of the screened and inverse-designed
materials, visualize ternary phase diagrams generated for many systems of
interest using ML predictions, and suggest strategies for further improvement
and expansion in the future.

††preprint: AIP/123-QED

## INTRODUCTION

The importance of engineering and optimizing halide perovskites (HaPs) for use
as absorbers in next-generation solar cells cannot be overstated. There is a
deluge of experimental and computational work on this topic emerging every
single day Ansari, Qurashi, and Nazeeruddin (2018); Yin  _et al._ (2015);
Manser, Christians, and Kamat (2016); Brenner  _et al._ (2016); Cui  _et al._
(2019); even if such attempts appear to be plateauing, incremental
improvements are important and motivate more comprehensive investigations.
Record power conversion efficiencies were recently reported by multiple
sources for perovskite-based tandem solar cells Website (2023a, b). Curiously,
there seems to be ever more room for better performance by tailoring the ABX3
composition (for canonical 3D crystalline perovskites) in terms of the number
and nature of species that inhabit the A/B/X sites, dopants, phase stability,
interfaces, and defects Yang and Mannodi-Kanakkithodi (2022). Of course, HaPs
may also adopt the double perovskite structure or a 2D phase via the use of
organic spacers, and be used in both single-junction or tandem solar cells
with wide or low band gap semiconductors.

Our research group has made several contributions to the HaP literature over
the last few years, reporting updates on discovering promising new materials
using density functional theory (DFT)-based computational screening and
machine learning (ML) Yang and Mannodi-Kanakkithodi (2022); Mannodi-
Kanakkithodi and Chan (2022); Yang, Manganaris, and Mannodi-Kanakkithodi
(2023); Yang and Mannodi-Kanakkithodi (2023). In 2022, we published a major
study on using DFT-ML to design novel B-site mixed ABX3 HaPs with desired
stability, band gap, photovoltaic (PV) figure of merit, and defect tolerance
Mannodi-Kanakkithodi and Chan (2022). In a review paper in 2022, we covered
several similar efforts from the literature utilizing high-throughput (HT)
computations and ML models trained on small, medium, or large datasets of
perovskite properties, to drive the discovery and understanding of HaPs Yang
and Mannodi-Kanakkithodi (2022). Recently, we published another comprehensive
study on generating one of the largest known DFT datasets of pseudo-cubic HaP
alloys, with mixing allowed at A, B, or X sites, from multiple DFT
functionals, resulting in (a) a thorough analysis of how common cation and
anion choices and types of mixing affect the stability and optoelectronic
properties, (b) an understanding of how different levels of theory in DFT
reproduce experimentally measured properties, and (c) an open-access dataset
that can be utilized by anybody for data mining and ML endeavors Yang,
Manganaris, and Mannodi-Kanakkithodi (2023). We followed up this study by
extending the DFT dataset to a series of non-cubic HaPs in multiple prototype
phases, using multiple semi-local and non-local DFT functionals, as well as
applying varying degrees of octahedral distortions and strain and different
types of ionic ordering in alloys Yang and Mannodi-Kanakkithodi (2023).

In the present contribution, we build upon our prior work and present multi-
fidelity ML regression models Chen  _et al._ (2021) trained on our existing
multi-phase multi-functional HaP dataset of $\sim$ 103 points or so, leading
to accurate predictions across hundreds of thousands of possible compounds,
screening of promising candidates, and inverse design using genetic algorithm
(GA) to expand the scope of materials selection beyond HT-screening. We create
a fusion dataset of HaPs containing three properties, namely the decomposition
energy, electronic band gap, and spectroscopic limited maximum efficiency
(SLME) Bercx  _et al._ (2016), estimated from two types of DFT functionals—the
semi-local GGA-PBE (570 data points), and non-local hybrid HSE06 with spin-
orbit coupling (SOC) (347 data points). This dataset of 917 DFT points is
further enhanced with 97 experimental data points collected from the
literature Almora  _et al._ (2021); Jacobsson (2022), reporting HaP
compositions and their measured band gap and photo-conversion efficiency
(PCE). Figure 1(a) shows a standard pseudo-cubic 2$\times$2$\times$2
perovskite supercell, and Figure 1(b) shows the chemical space considered in
our work in terms of A, B, and X species, with MA and FA representing organic
molecules methylammonium and formamidinium, respectively. Figure 1(c) further
shows the format of the dataset, with every compound represented in terms of a
14-dimensional composition vector (capturing fractions of the 5 A species, 6 B
species, and 3 X species in any compound), a 36-dimensional “elemental
properties" vector Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and
Mannodi-Kanakkithodi (2023) (12 weight-averaged properties each for A, B, and
X site species, such as ionic radius, electron affinity, etc.), and one of
four prototype perovskite phases it could adopt—cubic, tetragonal,
orthorhombic, or hexagonal. Additional columns represent the source of the
data (PBE, HSE, or experiment), and the properties of interest: decomposition
energy or $\Delta$H, band gap or Egap, and SLME (or PCE from experiment).

Figure 1:  (a) An example pseudo-cubic 2x2x2 HaP supercell. (b) The chemical
space of ABX3 HaPs studied in this work. (c) The perovskite dataset formatted
for multi-fidelity ML, including all inputs (composition, elemental
properties, phase, and fidelity) and outputs (decomposition energy, band gap,
and PV efficiency or SLME). (d) The perovskite design workflow showing data
formatting, regression optimization, prediction, GA, and screening steps.

The use of “multi-fidelity" learning Chen  _et al._ (2021); Mannodi-
Kanakkithodi  _et al._ (2020); Pilania, Gubernatis, and Lookman (2017) (or
multi-task learning Kuenneth  _et al._ (2021); Iraki  _et al._ (2023)) is
motivated by the fact that different DFT functionals work well for different
HaP compositions, and accuracy compared with experiments is not as trivial as
one would imagine. While GGA-PBE, including variants such as PBEsol (improved
PBE for solids Csonka  _et al._ (2009)) and PBE-D3 (explicit van der Waals
corrections Grimme  _et al._ (2010)), reproduces lattice parameters and
stability reasonably well, it is typically inaccurate for electronic, optical,
and defect properties Yang and Mannodi-Kanakkithodi (2022). Hybrid HSE06 is
much better for optoelectronic properties, but is expensive, especially when
SOC is incorporated for the relativistic effects of heavy atoms such as Pb
Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and Mannodi-
Kanakkithodi (2023). For hybrid organic-inorganic perovskites (HOIPs) such as
MAPbI3 and MAPbBr3, PBE without SOC often reproduces the experimental band gap
as well as HSE+SOC. While HSE+SOC should work for inorganic HaPs in general,
this is not always the case, as the mixing fraction $\alpha$ in HSE06 (default
value of $\alpha$ = 0.25) could itself be tuned; e.g., it was shown that for
CsPbI3, $\alpha$ = 0.41 reproduces the band gap more accurately Wang  _et al._
(2022), while $\alpha$ = 0.50 works best for cubic FAPbI3 Yang and Mannodi-
Kanakkithodi (2023). We posit that the true relationship between the
perovskite chemistry, DFT functional, and experimental properties, is
enormously complex, and by learning from a fusion dataset containing
properties from multiple levels of theory as well as from experiments,
comprehensive experiment-fidelity predictions could be achieved across a wide
chemical space. Such models would exploit inherent correlations between DFT
and experiments, as well as expand the reach of experimental predictions
beyond the range of chemistries for which measured data is currently
available, provided DFT data is indeed available in these unexplored spaces.

The remainder of this manuscript presents details of the ML approaches and
subsequent screening and inverse design, followed by a systematic discussion
of the results. The overall methodology is shown in Figure 1(d), going from
compiling the DFT+Expt dataset of properties and descriptors to training
several single- and multi-fidelity regression models, resulting in
enumeration, prediction and screening of promising candidates and inverse
design of new compounds using GA, closing the experiment-fidelity perovskite
design loop. We discuss the accuracy and merits of all single- and multi-
fidelity regression models based on random forest regression, and furthermore,
present an analysis of the best compounds from screening and GA, in terms of
the frequencies of occurrence of various chemical species and different types
of mixing at cation or anion sites. All data and models are openly available
to the community and will serve not only the discovery of novel HaPs for
optoelectronic applications, but also provide a fertile playground for testing
of a variety of ML techniques.

Figure 2:  The perovskite dataset divided in terms of source of data (DFT-PBE,
DFT-HSE, or experiment) and perovskite phase (cubic, tetragonal, orthorhombic,
and hexagonal). (a) Band gap plotted against decomposition energy. (b) PV
efficiency plotted against the band gap.

## METHODS

### Compiling the HaP Dataset

The entire HaP dataset is pictured in Figure 2, divided in terms of the source
of data (PBE, HSE, or Expt) and the perovskite phase (cubic, tetragonal,
orthorhombic, or hexagonal). This data is compiled from our recent
publications Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and
Mannodi-Kanakkithodi (2023); Yang and Mannodi-Kanakkithodi (2023). A majority
of the data is for the cubic phase as a result of our initial high-throughput
investigation Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and
Mannodi-Kanakkithodi (2023), whereas the non-cubic data was generated in
follow-up work Yang and Mannodi-Kanakkithodi (2023). 570 data points are from
PBE, 347 from HSE, and 97 from experiments Almora  _et al._ (2021); Jacobsson
(2022), resulting in a combined dataset of 1014 points. The decomposition
energy ($\Delta$H) shows how likely the ABX3 compound is to decompose to AX
and BX2 phases, with a mixing entropy contribution included as well, giving a
convenient per formula unit (p.f.u.) metric for perovskite stability Mannodi-
Kanakkithodi and Chan (2022); Yang, Manganaris, and Mannodi-Kanakkithodi
(2023). Egap comes from accurate PBE and HSE electronic structure computations
using dense k-point meshes, and from the experimental literature where
techniques ranging from UV-vis absorption to photoluminescence spectroscopy
have been applied Almora  _et al._ (2021); Jacobsson (2022). SLME is derived
at 5$\mu$m sample thickness from the DFT-computed optical absorption spectrum
based on previously developed approaches Bercx  _et al._ (2016); Choudhary
_et al._ (2019), and is combined with measured PCE values reported from solar
cells based on different HaP compositions. All DFT data are restricted to HaP
compositions with any A, B, or X constituent (shown in Figure 1(b)) occurring
only in fractions of n/8 (n = 0, 1, 2, … 8), such that geometry optimization
and subsequent electronic and optical calculations could be performed using
the special quasirandom structures (SQS) approach Jiang  _et al._ (2016) in
2$\times$2$\times$2 (cubic) or 2$\times$2$\times$1 (tetra, ortho, hex)
supercells. Other specific DFT details and data analysis can be found in past
publications Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and
Mannodi-Kanakkithodi (2023); Yang and Mannodi-Kanakkithodi (2023).

The Egap vs $\Delta$H plot in Figure 2(a) shows that while there are dozens of
compounds from all phases that lie in the $\Delta$H < 0.5 eV (a relaxed
threshold) and 1 eV < Egap < 2 eV range, a majority of the compounds,
primarily cubic, are in the undesirable ranges. Figure 2(b) shows that PV
efficiencies peak around Egap $\sim$ 1.5 eV and the favorable region (SLME/PCE
> 15%) is dominated by cubic compounds, though that might just be a factor of
the current dataset not containing as many non-cubic structures. From both
plots, it is evident that only 10 to 20% of the entire dataset of 1014 points
show desired stability and optoelectronic properties. The compounds in this
dataset cover the 14-dimensional ABX3 chemical space adequately (including 5 A
species, 6 B species, and 3 X species), in terms of how often and in what
mixing fraction any particular species appears in the compound Yang,
Manganaris, and Mannodi-Kanakkithodi (2023); Yang and Mannodi-Kanakkithodi
(2023). In reality, this space is practically infinite, as mixing fractions
could be as low or high as possible (and not just n/8 fractions), and any
number of ions could be mixed together to create high entropy perovskites.
Thus, a major motivation of this work is that learning from a modest dataset
of the order of 103 points, one could, in theory, make predictions for 105 to
106 points (or beyond) which would include all intermediate compositions and
mixing fractions missing from the original dataset, and perform more
comprehensive screening and design. Table 1 presents details of the PBE, HSE,
and experimental datasets in terms of the number of data points for each phase
and the available properties.

Data Fidelity | Total Data Points | Phases (Data Points) | Properties
---|---|---|---
PBE | 570 | Cubic (469), Tetra (34), Ortho (37), Hex (30) |  $\Delta$H, Eg, SLME
HSE | 347 | Cubic (246), Tetra (34), Ortho (37), Hex (30) |  $\Delta$H, Eg, SLME
Expt. | 97 | Phases assigned from DFT-ML Predictions | Eg, PCE
Table 1:  Description of the perovskite dataset used for training ML models in
this work, in terms of total number of data points, data points per perovskite
phase, and the properties available.

### Training Single- and Multi-Fidelity Surrogate Models

The dataset is formatted as shown in Figure 1(c) and fed into a random forest
regression (RFR) algorithm for rigorous training and optimization of
predictive models for each property. Standard data science and ML practices
are applied: an 80-20 train-test split is used for the entire dataset, 5-fold
cross-validation is applied, and grid-based hyperparameter optimization is
performed. Relevant modules are imported from the Scikit-learn library
Pedregosa  _et al._ (2011) and all our code is available on Github et al.
(2023). Four types of descriptors are used as input (X) to the RFR models for
predicting property Y ($\Delta$H, Egap, SLME/PCE):

  1. 1.

14-dimensional composition vector: This encodes the HaP composition in terms
of the fraction (between 0 and 1) of every A/B/X species in the compound.

  2. 2.

36-dimensional elemental properties vector: This uses 12 distinct well-known
properties (the full list is provided in Table S1 in the SI) to represent the
A, B, and X site constituents, using a weighted fraction when there is mixing
at any site.

  3. 3.

4-dimensional phase vector: This one-hot encodes whether the perovskite phase
is cubic, tetragonal, orthorhombic, or hexagonal.

  4. 4.

0-dim, 2-dim, or 3-dim fidelity vector: This one-hot encodes the source of the
data (PBE, HSE, or Expt) Chen  _et al._ (2021); Kuenneth  _et al._ (2021).

Three different types of RFR models are trained:

  1. 1.

PBE, HSE, and Expt. single-fidelity (SF) models: The source of the data is not
an input here, so a total of 54 dimensions are used as X and separate models
are trained to predict $\Delta$H, Egap and SLME each from PBE and HSE, as well
as Egap and PCE just from experiments. These models are subsequently referred
to as PBE-sf, HSE-sf, and Expt-sf, respectively.

  2. 2.

PBE+HSE multi-fidelity (MF) models: A 2-dim data source vector is included to
yield a 56-dimensional X, used as input to predict the three properties
simultaneously from both functionals based on the dataset of 917 DFT points.
The advantage here is that the reach and prediction accuracy of the smaller
and more expensive HSE dataset can be enhanced by utilizing its correlations
with the supposedly inferior but larger PBE dataset. These predictions will be
referred to as PBE-mf1 and HSE-mf1, respectively.

  3. 3.

PBE+HSE+Expt MF models: A 3-dim data source vector is included to yield a
57-dimensional X, which is used as input to predict Egap and SLME or PCE
simultaneously from PBE, HSE, and Expt based on the combined DFT-Expt dataset
of 1014 points. A clear advantage is that the applicability of the much
smaller experimental dataset can be enhanced by utilizing its correlations
with the PBE and HSE data. These predictions will be referred to as PBE-mf2,
HSE-mf2, and Expt-mf2, respectively.

RFR results after training and optimization are visualized in terms of parity
plots between ground truth values (actual PBE, HSE, or Expt) on the x-axis and
ML predictions on the y-axis, separately for each property and each source of
data. It should be noted that the ultimate test of the model’s predictive
power is its performance on unseen data points, and ideally, one must look at
the quality of prediction for every data point when it is considered as part
of the test set. To achieve this, we adopt the following ensemble-based
strategy: every SF and MF RFR model is trained 5000 times across each dataset,
with an 80-20 split, such that every single data point is considered in the
test set approximately 1000 times. Effective test set predictions are then
achieved for all points as a mean over their $\sim$ 1000 test predictions
(also yielding the standard deviation, which serves as a rough uncertainty in
prediction for any point). Thus, final parity plots, which will be presented
and discussed later, show only test predictions for all PBE, HSE, and Expt
points.

### Enumeration, Prediction, and Screening

Since only a small subset of possible ABX3 compositions is used for the DFT+
Expt dataset, we generate a much larger dataset of “hypothetical" HaP
compositions by populating the 14-dim composition vector with fractions of
n/8, such that the A components sum up to 1, B components sum up to 1, and X
components sum up to 3. A similar approach was applied in other works as well
Mannodi-Kanakkithodi and Chan (2022); Manganaris, Yang, and Mannodi-
Kanakkithodi (2023). To keep this combinatorial dataset somewhat tractable, we
restrict it to only one type of mixing at a time (i.e., there will not be
mixing at A site and B or X sites simultaneously) and only mixing fractions of
n/8. This leads to a total of 37,785 unique compositions including thousands
of A-site mixed, B-site mixed, and X-site mixed compounds each, which is
transformed to 151,140 data points considering them in four phases each. All
compounds are converted into descriptors and fed to the best RFR models,
eventually resulting in prediction of their PBE/HSE-fidelity $\Delta$H as well
as experiment-fidelity Egap and PV efficiency. Screening is performed to
obtain the subset of these > 150,000 compounds with low decomposition energy,
band gap between 1 and 2 eV, and PV efficiency > 15%.

### Inverse Design using GA

Finally, we move beyond the constraints of the restricted chemical space
considered for enumeration, to perform a more efficient design of HaP
compositions in any phase that satisfy conditions of DFT stability and
experiment-level optoelectronic properties. A constrained or clamped Genetic
Algorithm (GA) is used for this purpose Jennings  _et al._ (2019); Lourenço
_et al._ (2023); Kern  _et al._ (2021), using a procedure wherein hundreds of
arbitrary ABX3 compositions are generated, with any kind of mixing allowed at
A, B, or X sites, in any of the four phases, and a complex objective function
is defined for each compound taking the following factors into account:

  1. 1.

Chemical feasibility: Constituent fractions at A, B, and X sites should
respectively sum up to 1, 1, and 3.

  2. 2.

Suitability of modeling via DFT: Mixing fractions are restricted to be n/8,
n/27, or n/64 (where is a positive integer), such that any composition could
be simulated using a 2$\times$2$\times$2, 3$\times$3$\times$3, or
4$\times$4$\times$4 supercell.

  3. 3.

Stability: $\Delta$H should be minimized to maintain maximum likelihood of
perovskite formability. The HSE-level $\Delta$H prediction is used here.

  4. 4.

Optoelectronic properties: Egap should be in between 1 and 2 eV and PV
efficiency should be as high as possible, with both predictions at experiment-
fidelity.

During any GA run, new HaP compositions are generated using concepts such as
crossover, elitism, and mutation, so as to minimize the objective function,
yielding a plot between GA generation and objective function, eventually
resulting in the best material that satisfies all criteria. Running this
several hundred times leads to a massive list of attractive materials that
further improve on the compounds derived from enumeration $\rightarrow$
prediction $\rightarrow$ screening. Here, we run GA multiple times with
different types of inputs, such as restricting the phase to be cubic or
tetragonal, restricting the B-site to consider only Pb, Sn, and Ge, or to not
consider Pb at all in a bid to achieve Pb-free perovskites, etc. Final results
can be visualized for different families of HaPs in terms of PV efficiency vs
band gap plots for hundreds of stable compounds. The "geneticalgorithm2"
package is used for running the GA models et al. (2022).

## RESULTS AND DISCUSSION

### DFT-Only Single-Fidelity and Multi-Fidelity RFR Models

Figure 3:  Single-fidelity random forest regression models trained for
decomposition energy, band gap, and PV efficiency, separately on the DFT-PBE
and DFT-HSE datasets. Parity plots capture effective averaged test predictions
(over 5000 models) for every data point, and RMSE values are shown separately
for PBE and HSE.

Figure 3 shows the PBE-sf and HSE-sf models for $\Delta$H, Egap, and PV
efficiency, with PBE and HSE data points presented on the same plot. The
parity plots show effective test predictions, averaged over an ensemble of
5000 models, for all 917 data points from PBE and HSE. High accuracy is
achieved for both $\Delta$H and Egap, with RMSE values of 0.17 eV and 0.23 eV
respectively for PBE, and 0.20 eV and 0.30 eV respectively for HSE. These are
errors of 5% or less considering the range of values of both properties across
the PBE and HSE datasets, indicating a 95% accuracy. Given that there are
between 5 and 12 atoms p.f.u. in any HaP compound, the $\Delta$H RMSE converts
to $\sim$ 20-40 meV/atom, which are highly competitive with other ML formation
energy predictions in the literature Bartel  _et al._ (2020); Peterson and
Brgoch (2021). The same can be said for Egap prediction errors between 0.2 and
0.3 eV as well Zhuo, Mansouri Tehrani, and Brgoch (2018); Gao  _et al._
(2022). Predictions errors for PBE and HSE SLME are higher on average, with
RMSE values of 2.12% and 1.78% respectively; these errors are $\sim$ 10% of
the total range of values, and show that the composition-based descriptors may
not be sufficient for highly accurate prediction of PV efficiency. This is not
a surprise, as the optical absorption spectra are likely sensitive to the
perovskite structure and would thus affect the SLME prediction.
Furthermore—especially when experimental data will be included in
training—there are many factors beyond the perovskite composition alone,
including the solar cell architecture, interfaces, and external conditions,
that influence the PV efficiency. Nevertheless, the focus of current work is
on determining upper limit efficiencies as a function of HaP compositions and
eventually screening materials which are likely to show high efficiencies.

Figure 4:  Multi-fidelity random forest regression models trained for
decomposition energy, band gap, and PV efficiency, based on a combined DFT-PBE
+ DFT-HSE dataset. Parity plots capture effective averaged test predictions
(over 5000 models) for every data point, and RMSE values are shown for PBE and
HSE.

Next, the DFT-only MF models are trained for each property by adding 2
descriptors indicating whether the data source is PBE or HSE, leading to PBE-
mf1 and HSE-mf1 predictions. Results are presented in Figure 4, showing an
improvement in performance over the SF models, with models once again
picturing averaged test-only predictions over 5000 models. PBE and HSE
$\Delta$H models show RMSE values of 0.15 eV and 0.12 eV respectively. Egap
predictions also represent slight improvement, with PBE RMSE of 0.17 eV and
HSE RMSE of 0.20 eV, while SLME predictions show marked improvement with RMSE
of 1.78% for PBE and 1.35% for HSE. While $\Delta$H and Egap values in the HSE
dataset cover a wide range of values, it can be seen from Figures 2(b), 3, and
4 that the HSE SLME values clearly occur in a smaller range than the PBE SLME
values, which is an artifact of the nature of this dataset and has been
explained in prior work Yang, Manganaris, and Mannodi-Kanakkithodi (2023). As
such, we expect the MF models to theoretically expand the reach of the HSE
predictions to new regions in the chemical space that might show higher PV
efficiencies, something that may not be possible with SF models given the
interpolative nature of regression models. Generally, the PBE and HSE
$\Delta$H values, both SF and MF1 predictions, have a good correlation and can
be equally trusted, but given the low test prediction RMSE of 0.12 eV, the
HSE-mf1 $\Delta$H model should provide the best way for new prediction and
screening of the bulk stability of HaPs.

### RFR Models for Experiment-Fidelity Predictions

Finally, multiple approaches are used for making predictions on the
experimental data points, and to eventually achieve accurate and general
experiment-fidelity predictions. An important shortcoming of the dataset had
to be overcome here: while we could collect experimental data on nearly 100
points from the literature which included their chemical composition, band
gap, and PV efficiency (and some other solar cell-relevant quantities Almora
_et al._ (2021)), there was almost never sufficient information available
about the corresponding perovskite phase. An exhaustive search of every
composition in the experimental literature to determine its preferred phase is
beyond this scope of the current study, though it can and should be performed
in the future. Furthermore, many studies do not explicitly report the likely
phase of the material, though diffraction patterns and spectroscopic
measurements are reported, making it difficult to definitively assign phase
information to any compound, which makes a computational estimate a better
approach. As such, we apply the following strategy to create a 4-dimensional
phase vector (corresponding to cubic, tetragonal, orthorhombic, or hexagonal
phase) for every experimental data point:

  1. 1.

The PBE-sf $\Delta$H is predicted for all 100 compounds considered in all 4
phases.

  2. 2.

Predictions are made over an ensemble of 5000 models as described earlier, and
the most stable phase (which of the four phases has the lowest predicted PBE
$\Delta$H?) is noted each time.

  3. 3.

A score between 0 and 1 is assigned to each of the 4 dimensions of the phase
vector based on the relative frequency of occurrence of any phase as the most
stable. For instance, for the compound FASnI3, the cubic phase fraction is
found to be 0.57 and the hexagonal phase fraction is 0.41.

  4. 4.

With all 54 dimensions (including composition, phase, and elemental property
vectors) now available for all experimental data points, single-fidelity
models could be trained, and this data can be added to the DFT data along with
a 3-dimensional fidelity vector to train the new MF models.

Figure 5:  DFT-ML predictions compared to experimental values for (a) single-
fidelity PBE and HSE models, and (b) DFT-only multi-fidelity models. Expt vs
DFT RMSE values are shown separately for PBE and HSE. Figure 6:  Experiment-
only single-fidelity RFR predictions for (a) band gap, and (b) PV efficiency,
with effective test predictions and corresponding RMSE shown.

In the absence of any experimental data, one might wonder how the DFT-
predicted Egap and SLME compare with measured values. A comparison of DFT vs
experiments has been performed for a small set of compounds in past work, and
the general understanding is that PBE and HSE errors are quite large for a lot
of compounds Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and
Mannodi-Kanakkithodi (2023); Yang and Mannodi-Kanakkithodi (2023). To examine
this effect further, we used the PBE-sf, HSE-sf, PBE-mf1, and HSE-mf1 models
to make predictions on all 97 experimental data points, using the
54-dimensional inputs described above. The DFT-ML SF and MF predictions of
Egap are plotted against actual experimental values in Figure 5(a) and (b)
respectively: it can be seen that PBE-sf predictions have the lowest RMSE of
0.4 eV, with all other predictions lying in the $\sim$ 0.6 eV RMSE range.
These errors are clearly too high, and certain compounds are especially poorly
predicted, with errors larger than 1.5 eV at times. Corresponding parity plots
comparing PBE-sf, HSE-sf, PBE-mf1, and HSE-mf1 predictions of SLME with
experimental PCE are presented in Figure S1, showing that there is essential
no correlation between the DFT-ML and experimental values of PV efficiency.

Next, SF models were trained for the experimental Egap and PCE, and effective
test predictions for all 97 points based on an ensemble of 5000 models are
presented in Figure 6. Egap predictions show a very low RMSE of 0.11 eV,
though a caveat is the lower range of values in the experimental dataset,
whereas RMSE on PCE predictions is much higher at 3.72%. Although the Expt-sf
models could already be used for making experiment-fidelity predictions on
hundreds of thousands of hypothetical compounds, it would be limited and
erroneous given the narrow chemical space in the experimental dataset compared
to the DFT dataset—where explicit care was taken to involve all chemical
species in roughly equal proportions across all the compounds. Thus, a DFT-
Expt MF model might serve high-throughput predictions better.

PBE-mf2, HSE-mf2, and Expt-mf2 prediction performances are pictured in Figure
7. PBE and HSE Egap predictions show similar RMSE of 0.17 eV and 0.19 eV
respectively, similar to the MF1 models. The same is true for SLME predictions
where the PBE RMSE is 1.72% and the HSE RMSE is 1.29%. Expt-mf2 Egap RMSE is
0.14 eV and PCE RMSE is 3.62%. These predictions are again quite similar to
the Expt-sf performances, but are vastly more useful for their applicability
across new regions of the HaP chemical space. We believe that the MF2 models
learn inherent relationships between Expt, PBE, and HSE values, and are thus
capable of making experiment-fidelity predictions for chemistries that have
been studied from PBE and/or HSE. Table 2 lists the test RMSE values for all
three properties from many different RFR models, with the values in bold
showing the lowest errors for any specific property at any fidelity.

Figure 7:  Multi-fidelity random forest regression models trained for the band
gap and PV efficiency from DFT-PBE, DFT-HSE, and experiment. Parity plots
capture effective test predictions averaged over 5000 models, and RMSE values
are shown separately for PBE, HSE, and Expt.

### Understanding Feature Importance

Figure S2 shows the Pearson coefficient of linear correlation Benesty  _et
al._ (2009) plotted between the 56-dimensional vector representing every
compound in the DFT-mf1 dataset (PBE + HSE data), against the three properties
of interest. The primary observations here are very similar to past
publications Mannodi-Kanakkithodi and Chan (2022); Yang, Manganaris, and
Mannodi-Kanakkithodi (2023); Yang and Mannodi-Kanakkithodi (2023). Increasing
the K fraction will lead to an increase in $\Delta$H and thus make the
compound less stable, while the FA fraction has the exact opposite effect.
$\Delta$H is highly negatively correlated with features such as the ionic
radius and atomic number of the A-site cation, showing that larger cations (to
a point) are best for perovskite stability. While Ca, Sr, and Ba lead to a
heavy increase in Egap, it is very negatively correlated with the electron
affinity, ionization energy, and electronegativity of the B-site cations. The
SLME shows the opposite trends to what is found for Egap, with increase in the
B-site electron affinity, ionization energy, and electronegativity leading to
an increase in efficiency. Interestingly, we generally find low correlation
between the phase/fidelity and the properties, except for the positive
correlation between the cubic phase and both the $\Delta$H and the Egap, which
is likely a consequence of the dominance of cubic structures in the dataset
that means a majority of the unstable compounds as well as a majority of the
wide band gap compounds are cubic.

To understand the contributions of various features better, we plotted the
feature importance values for all descriptor dimensions as obtained from the
best RFR models, in Figure S3. It can be seen that the main contributions to
$\Delta$H are from the A and X species, as well as from the elemental
properties that highlight the size of the A-site cations and X-site
anions—similar to the observations from Figure S2. For Egap and PV efficiency,
there is an obvious dominance from the B-site electronegativity followed by
the electron affinity and ionization energy, showing that the optoelectronic
properties are heavily determined by the B-site cations (and a little bit from
the X-site anions) and have almost no sensitivity to the A-site species.
Figure S4 shows the Egap and PV efficiency for the entire dataset of 1014
points plotted against the B-site electronegativity, showing a rough
correlation, especially for the DFT data.

RFR Model |  $\Delta$H RMSE (eV) | Egap RMSE (eV) | PV Efficiency RMSE (%)
---|---|---|---
PBE-sf | 0.17 | 0.23 | 2.12
HSE-sf | 0.20 | 0.30 | 1.78
Expt-sf | - | 0.11 | 3.65
PBE-sf vs Expt. | - | 0.40 | 5.45
HSE-sf vs Expt. | - | 0.60 | 5.48
HSE-sf vs Expt., shifted | - | 0.31 | -
PBE-mf1 | 0.15 | 0.17 | 1.78
HSE-mf1 | 0.12 | 0.20 | 1.35
PBE-mf1 vs Expt. | - | 0.59 | 5.28
HSE-mf1 vs Expt. | - | 0.62 | 5.50
PBE-mf1 vs Expt., shifted | - | 0.34 | -
HSE-mf1 vs Expt., shifted | - | 0.36 | -
PBE-mf2 | - | 0.17 | 1.72
HSE-mf2 | - | 0.19 | 1.29
Expt-mf2 | - | 0.09 | 2.24
Table 2:  Root mean square error (RMSE) in test set prediction for the three
properties from multiple single- and multi-fidelity models. Figure 8:  A
glimpse of high-fidelity predictions made over $\sim$ 150,000 HaPs: (a)
Predicted band gap (from Expt-mf2) plotted against predicted decomposition
energy (from HSE-mf1) for cubic compounds, and (b) predicted PV efficiency
(from Expt-mf2) plotted against predicted band gap for tetragonal phase
compounds. Figure 9:  Frequencies of occurrence of the 14 chemical species
within the list of 3610 promising compounds obtained via enumeration,
prediction, and screening.

### Prediction and Screening at Experiment-Fidelity

Next, predictions are made for the expanded set of 151,140 compounds populated
across the same HaP chemical space. It should be noted that for any given
compound, PBE or HSE computations would at least require at least 12 to 24
hours of CPU time (on approximately 128 cores) for estimating the three
properties, which would correspond to over 200 years of computing time for >
150,000 compounds. Experimental investigation of all these materials would
take even longer. The ML predictions, on the other hand, require only seconds
for several compounds, and can be made for hundreds of thousands of compounds
in mere minutes. Figure 8 along with Figures S5 and S6 show the HSE-mf1
$\Delta$H, Expt-mf2 Egap, and Expt-mf2 PCE predicted for the 37,785 unique
compositions in four different phases, as $\Delta$H vs Egap and Egap vs PV
efficiency plots.

It can immediately be seen that (i) general shape and distribution of the
plots is the same as the DFT-Expt dataset pictured in Figure 2, and (ii) there
are now hundreds of more compounds that exist in the desirable ranges of
$\Delta$H < 0.2 eV, Egap between 1 and 2 eV, and PCE > 15%. We find that 3610
compounds out of the 151,140 fulfil each of these criteria, which represents
2.4% of the entire population. 1180 of these compounds are cubic, 856
tetragonal, 889 orthorhombic, and 685 hexagonal. Further, a majority of these
compounds are B-site mixed (2036 in numbers) while 1364 compounds are A-site
mixed, and only about 200 compounds have halogen mixing. Figures S5 and S6
further show that the shapes of the plots look generally similar for different
phases, and there are some empty regions which will likely be covered by other
intermediate compositions we have not considered in this enumeration exercise.
Thus, we are able to use the multi-fidelity DFT-Expt predictions to screen >
3600 compounds with multiple desired properties. A further examination of the
chemical space of the screened compounds, presented in Figure 9, shows that FA
followed by Cs and MA are the most prevalent A-site cations, Pb, Ge, and Sn
are most common at the B-site, while nearly all the compounds are Iodides.
This is consistent with the most typical HaP compositions used in high
efficiency solar cells arising from FA-MA-Cs mixing at the A-site and a
predominance of Pb and I at the other sites. Table 3 shows the total number of
screened compounds divided in terms of the predicted perovskite phase and type
of mixing (pure, A-mixed, B-mixed, or X-mixed).

### Discovery at Experiment-Fidelity using GA

The GA results are presented in Figure 10, where part (a) shows a typical GA
run in the form of the objective function being minimized plotted against the
generation number (a maximum of 100 generations is run each time), and part
(b) shows the PV efficiency plotted against Egap for compounds obtained from
nearly 3000 different GA runs. It should be noted here that the GA models
utilize the HSE-mf1 prediction for $\Delta$H and Expt-mf2 predictions for Egap
and PCE, and mixing is allowed to happen in many different fractions, often
simultaneously at A, B, and X sites. Every GA run requires 2 to 5 minutes on
average for the 14-dimensional composition space with pre-trained RFR models
for property prediction. We run GA models for four different subsets of the
chemical space (D1, D2, D3, and D4), and show results for them in Figure
10(b). These datasets have been explained in terms of their constituent
chemical species in Table 4; e.g., D2 represents Pb-free perovskites whereas
D4 represents the much smaller space of (Cs-MA-FA)PbI3 perovskites.

Figure 10:  (a) An example GA run presented in terms of the objective function
plotted against the number of generations, showing the property and chemical
feasibility constraints. (b) Predicted (experiment-fidelity) PV efficiency
plotted against band gap for $\sim$ 3000 compounds from several GA runs; all
compositions are chemically meaningful and have decomposition energy < 0.2 eV
p.f.u.

Since many different constraints are applied in the objective function, the
best compounds obtained from a GA run may not be completely stoichiometrically
accurate or meet every property target. As can be seen in Figure 10(b), while
all compounds essentially fulfill the Egap requirement (as well as the
stability requirement, which is not plotted here), PCE is not always higher
than 15%. We find that across all the GA runs, there are a total of 1703
chemically meaningful ABX3 alloys which satisfy conditions for all three
properties; 484 of these compounds are cubic, 428 are tetragonal, 431 are
orthorhombic, and 360 are hexagonal phase compounds. We also find that the
total number of promising compounds increases when the chemical space becomes
less complex—that is, a 5-dimensional space given by Cs-MA-FA-Pb-I leads to
many more compounds with the targeted properties than the 14-dimensional space
of K-Rb-Cs-MA-FA-Ca-Sr-Ba-Ge-Sn-Pb-I-Br-Cl. This is both a consequence of the
difficulty of optimization in a high-dimensional space and the ease of finding
attractive materials in the Cs-MA-FA-Pb-I space which are already among the
best for stability and high efficiency. Figure 10(b) further shows that there
are scores of Pb-free HaPs (D2) with high PV efficiencies, which opens up an
important avenue towards eliminating Pb from solar cell perovskites. Many of
the compounds are 14-dim, which means they are allowed to contain any species,
while the remaining compounds are restricted to contain the most common
species found in the experimental literature, namely Cs, MA, and FA at A-site,
Pb, Sn, and Ge at the B-site, and Br/I at the X-site.

An examination of the frequencies of occurrence in the GA-screened list of
1703 compounds is presented for the chemical spaces D1, D2, D3, and D4 in
Figure 11, with the results somewhat more interesting than the compounds
screened from pure enumeration and prediction. In D1 and D2, we find FA and MA
are the most prevalent at the A-site, followed by Cs, with Rb and K clearly
preferred in smaller mixing fractions. Sn, Ge, and Pb are again the most
common B-site cations, and Br makes a considerable appearance at X although I
remains the king. In the total absence of Pb, Sn and Ge majorly dominate the
B-site. Figure 11(a) and (b) reveal that the best combination of properties
could potentially be achieved using some combination of FA and MA at the
A-site with small quantity of Cs, Sn and Ge at the B-site, and I with
potentially small quantities of Br or Cl at the X-site. Further, the lower-
dimensional chemical spaces D3 and D4 in Figure 11(c) and (d) respectively
show a preference for FA, Pb, and I, with Cs and Br being useful in smaller
fractions, but Sn and Ge still holding their own.

Phase | All Screened Compounds | Pure | A-mixed | B-mixed | X-mixed
---|---|---|---|---|---
All | 3610 | 9 | 1364 | 2036 | 201
Cubic | 1180 | 3 | 473 | 626 | 78
Tetra | 856 | 2 | 290 | 510 | 54
Ortho | 889 | 2 | 361 | 470 | 56
Hex | 685 | 2 | 240 | 430 | 13
Table 3: Number of ML-screened compounds with all properties in desirable ranges, namely $\Delta$H (HSE-mf1) < 0.2 eV, Egap (Expt-mf2) between 1 and 2 eV, and PCE (Expt-mf2) > 15%. The screened compounds are divided based on their predicted phase and type of mixing. Chemical Space | Description | Screened Compounds | Cubic | Tetragonal | Orthorhombic | Hexagonal
---|---|---|---|---|---|---
D1 | (K-Rb-Cs-MA-FA)(Ca-Sr-Ba-Ge-Sn-Pb)(I-Br-Cl)3 | 226 | 80 | 51 | 58 | 37
D2 | Pb-free, (K-Rb-Cs-MA-FA)(Ca-Sr-Ba-Ge-Sn)(I-Br-Cl)3 | 297 | 90 | 70 | 71 | 66
D3 | (Cs-MA-FA)(Ge-Sn-Pb)(I-Br)3 | 381 | 114 | 107 | 102 | 58
D4 | (Cs-MA-FA)PbI3 | 799 | 200 | 200 | 200 | 199
Table 4: The number of GA-screened compounds obtained for different chemical
sub-spaces (D1, D2, D3, and D4), divided in terms of the perovskite phase.
Figure 11:  Frequencies of occurrence of the 14 chemical species within the
list of 1703 promising compounds obtained from GA, for subsets of the chemical
space: (a) D1, (b) D2, (c) D3, and (d) D4. Figure 12:  Ternary property
diagrams showing the band gap and PV efficiency from Expt-mf2 predictions,
plotted for the chemical spaces Cs(Pb-Sn-Ge)I3, FA(Pb-Sn-Ge)I3, and MA(Pb-Sn-
Ge)I3.

### Visualizing Composition-Property Spaces using ML Predictions

Using ML predictions made across the set of 151,140 HaP compounds in different
phases, it is now possible to visualize how the phase stability and computed
properties vary in specific families of compounds, such as with increasing Sn
concentration in Pb-Sn mixed compounds. Here, we pick a few interesting and
important chemical spaces and visualize their property variations using
ternary diagrams. Figure 12 shows the Expt-mf2 Egap and Expt-mf2 PV efficiency
for the chemical spaces Cs(Ge-Sn-Pb)I3, MA(Ge-Sn-Pb)I3, and FA(Ge-Sn-Pb)I3,
plotted as ternary color maps with Ge, Sn, and Pb representing the three sides
of the triangle. It can be seen that in the Cs(Ge-Sn-Pb)I3 series, maximum
Egap > 1.5 eV is shown by Pb-dominated compositions, minimum Egap $\sim$ 1.1
eV by Sn-dominated compositions, whereas Ge generally leads to more
intermediate Egap values. MA(Ge-Sn-Pb)I3 compounds show very similar Egap
trends where in FA(Ge-Sn-Pb)I3 compounds, a dominance of Pb or Ge leads to
higher Egap values in the $\sim$ 1.8 eV range. The highest PV efficiencies are
obtained by majority Pb compositions with some Sn and Ge added in all three
series of compounds, leading to values > 12% for MA(Ge-Sn-Pb)I3, > 15% for
Cs(Ge-Sn-Pb)I3, and $\sim$ 18% for FA(Ge-Sn-Pb)I3. It can also be seen that
majority Ge compositions in FA(Ge-Sn-Pb)I3 lead to much lower PV efficiencies
around 13%, so Pb-based compositions may still be crucial to achieve high
efficiencies.

Figures S7 to S12 further show ternary plots for the phase stability (cubic vs
tetra vs ortho vs hex), the HSE-mf1 $\Delta$H, and the Expt-mf2 Egap and PV
efficiency, across the chemical spaces Cs(Ge-Sn-Pb)I3, MA(Ge-Sn-Pb)I3, FA(Ge-
Sn-Pb)I3, Cs(Ge-Sn-Pb)Br3, MA(Ge-Sn-Pb)Br3, and FA(Ge-Sn-Pb)Br3. Such diagrams
could be trivially plotted for any binary, ternary, or even higher-dimensional
chemical spaces of interest, and can be navigated to determine the likely
phase and bulk stability of any composition as well as their optoelectronic
properties. It can be seen that the Cs-iodides generally prefer the tetragonal
phase while the Cs-bromides prefer the orthorhombic phase. Nearly all
compositions are stable w.r.t. decomposition in either series, and the highest
PV efficiencies $\sim$ 13% for the bromides are achieved with Sn-dominance at
the B-site. The MA-based iodides and bromides are nearly always tetragonal and
show good bulk stability. Maximum efficiencies are obtained from mixed
compounds with intermediate Pb fractions. Finally, the FA-based compounds are
almost always stable in the hexagonal phase, except for a small area of cubic
stability in the iodides. Sn-dominant FA-bromides provide a peak PV efficiency
around 10%, but the real pay-off comes from Pb-dominant iodides. The entire FA
HaP space is highly stable against decomposition with $\Delta$H values < -1 eV
p.f.u.

## CONCLUSIONS

In summary, several single-fidelity and multi-fidelity random forest
regression models were trained for predicting the decomposition energy, band
gap, and photovoltaic efficiency of halide perovskites at both DFT and
experiment fidelity, resulting in the discovery of hundreds of promising new
candidates via high-throughput screening and genetic algorithm. This work is
based on an innovative approach of fusing experimental data from the
literature with multiple types of DFT computed properties, and the unique
representation of every material in terms of its composition and phase, well-
known elemental or molecular properties of species at the cation and anion
sites, and one-hot encoding fidelity information. All DFT data, optimized RFR
models, and predictions made on hundreds of thousands of hypothetical
compounds, are openly available to the community. GA enabled the efficient
design of compositions beyond the limits of brute-force enumeration in certain
mixing fractions, and provides an avenue for continuous discovery and
improvement with infusion of more data and expanding the chemical space.
Composition-property distributions are visualized in terms of ternary diagrams
generated using ML predictions, revealing the most promising regions of B-site
mixed (with Pb, Sn, and Ge) Cs/MA/FA iodides and bromides, and such exercises
could easily be repeated for any chemical sub-spaces of interest. Limitations
of the current work arise in terms of the experimental data only covering a
small range of band gaps, any compound only represented as a single data point
although it may posses dozens of polymorphs with varying properties, known
issues with the DFT functionals, and the lack of consideration of other
essential properties such as defect behavior and electron/hole mobilities.
Each of these factors will be explored in future work; some of our ongoing
work includes additional benchmarking of functionals for HaPs and the use of
crystal graph-based neural networks for property prediction based on entire
crystal structures as input. We anticipate that our datasets will serve many
data mining and ML efforts, both within our group and from the community in
general, and significant improvements will further be made in terms of data
and ML approaches.

## Conflicts of Interest

There are no conflicts to declare.

## Data Availability

Tabulated data is included as spreadsheets in the supporting documents. All
DFT data and ML models, including python scripts in Jupyter notebooks, can be
found on Github: https://github.com/mannodiarun/perovs$\\_$mfml$\\_$ga.

## Acknowledgements

This work was performed at Purdue University, under startup account
F.10023800.05.002 from the Materials Engineering department. This research
used resources of the National Energy Research Scientific Computing Center
(NERSC), the Laboratory Computing Resource Center (LCRC) at Argonne National
Laboratory, and the Rosen Center for Advanced Computing (RCAC) clusters at
Purdue.

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Supplemental material to "Discovering Novel Halide Perovskite Alloys using
Multi-Fidelity Machine Learning and Genetic Algorithm"
Jiaqi Yang,1, Panayotis Manganaris,1 and Arun Mannodi-Kanakkithodi,1

1School of Materials Engineering, Purdue University, West Lafayette, Indiana
47907, USA

Descriptor Label | Descriptor Meaning
---|---
K_frac | K fraction over A sites
Rb_frac | Rb fraction over A sites
Cs_frac | Cs fraction over A sites
MA_frac | MA fraction over A sites
FA_frac | FA fraction over A sites
Ca_frac | Ca fraction over B sites
Sr_frac | Sr fraction over B sites
Ba_frac | Ba fraction over B sites
Ge_frac | Ge fraction over B sites
Sn_frac | Sn fraction over B sites
Pb_frac | Pb fraction over B sites
Cl_frac | Cl fraction over X sites
Br_frac | Br fraction over X sites
I_frac | I fraction over X sites
Cubic | 1 if cubic, 0 if not
Tetra | 1 if tetragonal, 0 if not
Ortho | 1 if orthorhombic, 0 if not
Hex | 1 if hexagonal, 0 if not
PBE | 1 if data is from PBE, 0 if not
HSE | 1 if data is from HSE, 0 if not
Expt | 1 if data is from Expt, 0 if not
A_ion_rad | Ion radius of A site elements
A_BP | Boiling point of A site elements
A_MP | Melting point of A site elements
A_dens | Density of A site elements
A_at_wt | Atomic weight of A site elements
A_EA | Electron affinity of A site elements
A_IE | Ionization energy of A site elements
A_hof | Heat of formation of A site elements
A_hov | Heat of vaporization of A site elements
A_En | Electronegativity of A site elements
A_at_num | Atomic number of A site elements
A_period | Period number of of A site elements
B_ion_rad | Ion radius of B site elements
B_BP | Boiling point of B site elements
B_MP | Melting point of B site elements
B_dens | Density of B site elements
B_at_wt | Atomic weight of B site elements
B_EA | Electron affinity of B site elements
B_IE | Ionization energy of B site elements
B_hof | Heat of formation of B site elements
B_hov | Heat of vaporization of B site elements
B_En | Electronegativity of B site elements
B_at_num | Atomic number of B site elements
B_period | Period number of of B site elements
X_ion_rad | Ion radius of X site elements
X_BP | Boiling point of X site elements
X_MP | Melting point of X site elements
X_dens | Density of X site elements
X_at_wt | Atomic weight of X site elements
X_EA | Electron affinity of X site elements
X_IE | Ionization energy of X site elements
X_hof | Heat of formation of X site elements
X_hov | Heat of vaporization of X site elements
X_En | Electronegativity of X site elements
X_at_num | Atomic number of X site elements
X_period | Period number of of X site elements
Table SI:  Explanation of every descriptor dimension used in this work,
enabling the development of RFR models and subsequent predictions on new
materials. Figure S1:  DFT-ML SLME plotted against experimental PCE for (a)
single-fidelity RFR model, and (b) multi-fidelity RFR model. Figure S2:
Pearson coefficients of linear correlation between the 56-dimensional
descriptors (for the PBE+HSE dataset) and the three properties computed from
PBE and HSE. Figure S3:  Importance values of each feature from multi-fidelity
random forest regression trained on the combined PBE, HSE, and Expt dataset,
for (a) decomposition energy, (b) band gap, and (c) PV efficiency. Figure S4:
PBE, HSE, and expt. properties plotted against B-site electronegativity: (a)
band gap, and (c) PV efficiency. Figure S5:  Visualization of the ML-predicted
properties across the expanded chemical space of $\sim$ 150,000 compounds, in
terms of band gap (Expt-mf2) vs decomposition energy (HSE-mf1) plots. Figure
S6:  Visualization of the ML-predicted properties across the expanded chemical
space of $\sim$ 150,000 compounds, in terms of PV efficiency (Expt-mf2) vs
band gap (Expt-mf2) plots. Figure S7:  Ternary property diagrams showing the
predicted phase stability, decomposition energy, band gap, and PV efficiency
for Cs(Ge-Sn-Pb)I3 perovskites. Figure S8:  Ternary property diagrams showing
the predicted phase stability, decomposition energy, band gap, and PV
efficiency for Cs(Ge-Sn-Pb)Br3 perovskites. Figure S9:  Ternary property
diagrams showing the predicted phase stability, decomposition energy, band
gap, and PV efficiency for MA(Ge-Sn-Pb)I3 perovskites. Figure S10:  Ternary
property diagrams showing the predicted phase stability, decomposition energy,
band gap, and PV efficiency for MA(Ge-Sn-Pb)Br3 perovskites. Figure S11:
Ternary property diagrams showing the predicted phase stability, decomposition
energy, band gap, and PV efficiency for FA(Ge-Sn-Pb)I3 perovskites. Figure
S12:  Ternary property diagrams showing the predicted phase stability,
decomposition energy, band gap, and PV efficiency for FA(Ge-Sn-Pb)Br3
perovskites.