# Inorganic synthesis-structure maps in zeolites with machine learning and
crystallographic distances

Daniel Schwalbe-Koda  Lawrence Livermore National Laboratory, Livermore, CA,
United States<EMAIL_ADDRESS>Daniel E. Widdowson  University of Liverpool,
Liverpool, United Kingdom  Tuan Anh Pham  Lawrence Livermore National
Laboratory, Livermore, CA, United States  Vitaliy A. Kurlin  University of
Liverpool, Liverpool, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Zeolites are inorganic materials known for their diversity of applications,
synthesis conditions, and resulting polymorphs. Although their synthesis is
controlled both by inorganic and organic synthesis conditions, computational
studies of zeolite synthesis have focused mostly on organic template design.
In this work, we use a strong distance metric between crystal structures and
machine learning (ML) to create inorganic synthesis maps in zeolites. Starting
with 253 known zeolites, we show how the continuous distances between
frameworks reproduce inorganic synthesis conditions from the literature
without using labels such as building units. An unsupervised learning analysis
shows that neighboring zeolites according to our metric often share similar
inorganic synthesis conditions, even in template-based routes. In combination
with ML classifiers, we find synthesis-structure relationships for 14 common
inorganic conditions in zeolites, namely Al, B, Be, Ca, Co, F, Ga, Ge, K, Mg,
Na, P, Si, and Zn. By explaining the model predictions, we demonstrate how
(dis)similarities towards known structures can be used as features for the
synthesis space. Finally, we show how these methods can be used to predict
inorganic synthesis conditions for unrealized frameworks in hypothetical
databases and interpret the outcomes by extracting local structural patterns
from zeolites. In combination with template design, this work can accelerate
the exploration of the space of synthesis conditions for zeolites.

##  1 Introduction

Zeolites are inorganic porous materials widely recognized for their rich
polymorphism and numerous applications.1, 2, 3 Their porous structure provides
unique opportunities to tailor materials performance in catalysis, gas
adsorption, selective membranes, and more.4, 5, 6 In principle, the
performance of zeolites for each application can be controlled by adequate
selection of polymorph and composition. However, this selection is often
hindered by the high-dimensional synthesis routes required to produce the
materials 7. Zeolites are often synthesized with hydrothermal treatments, with
inorganic and organic precursors cooperating to crystallize the nanoporous
structure 8. Organic templates are known to direct the formation of certain
topologies, thus biasing the phase competition landscape to favor the best-
matching topology instead of another 9, 8. Because of this effect, design of
organic structure-directing agents (OSDAs) led to multiple successful examples
of phase-selective zeolite synthesis and control of catalytic properties 10,
11, 12, especially when used in combination with computational methods 13, 14,
15, 16, 17, 18, 19.

On the other hand, computational design of inorganic synthesis conditions for
zeolites has not yet achieved the same impact as template design. Despite
their promise in controlling active site distribution 20, phase selectivity
21, Si/Al ratio 22, morphology 23, or lowering the cost of syntheses 24,
selection of inorganic conditions capable of synthesizing existing and novel
zeolites is not easily modeled 25. Recent progress in quantifying the role of
inorganic synthesis conditions in zeolites includes: coupling machine learning
and literature extraction 26, 27; obtaining structure-synthesis correlations
from synthesis routes 21, 28; predicting effects of inorganic cations in
heteroatom distributions 20, 17; or using ML to control composition and
particle sizes from template-free syntheses 22. Nevertheless, their reliance
on reported data prevents them to propose inorganic conditions for the
synthesis of novel or hypothetical frameworks. Whereas some inorganic
synthesis-structure relationships can be derived from building units 29, 30,
27 or alternative structural descriptors 28, automatically screening for new
structures in hypothetical zeolite databases requires bypassing human-crafted
labels such as building units. Furthermore, although graph-theoretical methods
can detect composite building units (CBUs) in arbitrary structures, their
computational cost may be prohibitive when exploring large datasets. Data-
driven methods based in the topology of the structure also provide information
on key factors that govern kinetics of zeolite crystallization 31, 32, but do
not immediately inform their synthesis conditions. Finally, aggregate
framework information such as density-energy plots 33, 34 or local interatomic
distances 35 provide few correlations between different inorganic synthesis
conditions and targeted frameworks, which motivate new data-driven approaches
to synthesizability prediction 36. Thus, advancing towards a priori discovery
of novel zeolite frameworks requires developing methods to: (1) uncover
synthesis-structure relationships in zeolites; (2) efficiently explore the
inorganic synthesis space of zeolites; and (3) bypass the absence of labeled
data in hypothetical zeolite databases.

In this work, we correlate inorganic synthesis conditions to zeolite structure
using a mathematically strong representation for comparing periodic crystals,
the Average Minimum Distance (AMD)37, derived from the Pointwise Distance
Distribution (PDD)38 of crystal structures (see Fig. 1). The PDD is
independent of a unit cell, continuous under small perturbations,
theoretically complete for generic crystals, and distinguished all periodic
crystals in the Cambridge Structural Database. Importantly, it can be computed
much more efficiently than graph-based approaches, only requiring a fast
nearest neighbor search 39. We show that the AMD vectors of zeolites can be
used to predict inorganic synthesis conditions and recall a comprehensive
dataset of synthesis conditions from the literature. Then, we demonstrate that
unsupervised and supervised machine learning (ML) methods can be used to
create structure-synthesis relationships independently from OSDA design.
Finally, we propose inorganic synthesis conditions to realize hypothetical
frameworks based on distances toward structures whose synthesis is known, thus
proposing interpretable synthesis-structure models to guide the synthesis of
new zeolites.

Figure 1: Computational methods used to extract relationships between zeolite
structures and their associated inorganic synthesis conditions. a, Using the
concept of AMD and the distance between these invariants, we compute a
distance matrix between known zeolites. b, This information is combined with
literature data and ML methods to correlate structural patterns with inorganic
conditions.

##  2 Results and Discussion

###  2.1 Inorganic synthesis maps from unsupervised learning

Designing synthesis-structure relationships in zeolites has long relied on
intuitive patterns emerging from their natural structure. For example, some
CBUs are typically known to be synthesized by different inorganic conditions,
such as d6r in the presence of sodium ions or d4r in the presence of germanium
or fluorine. Nevertheless, not all structures produced with certain inorganic
agents exhibit the same CBUs, and CBUs are not necessarily realized only by
one element. Data-driven methods showed promise in connecting zeolite
synthesizability to their local structure 35 or accelerating their screening
36, but interpreting large databases of structures can be challenging
depending on the selected data representations. For instance, subgraph
isomorphism used to extract building units is an NP-type problem, and requires
creating a predefined set of substructures that are being searched for. On the
other hand, local structural fingerprints such as the Smooth Overlap of Atomic
Positions (SOAP) 40 may be unable to distinguish between certain atomic
environments 41.

To address these problems, we created a procedure to compare zeolites and
extract synthesis-structure relationships without relying on any structural
labels except for the atomic positions (Fig. 1). Using the concept of PDDs and
AMDs, we calculated the distance between two zeolite structures by comparing
their AMD (see Methods). This comparison between average local environments is
computationally efficient but descriptive enough to distinguish all crystals
in the CSD 38. Then, we assumed that zeolites sharing similar local structures
exhibit similar inorganic synthesis conditions. To test this hypothesis, we
computed the distance matrix between 253 known frameworks in the International
Zeolite Association (IZA) database using AMD, and then performed a qualitative
analysis of the results. We found that the AMDs correlated weakly with
differences of density and with the SOAP distance between structures, but
showed almost no correlation with graph-based distances from previous work 42
(Fig. S1). Moreover, we noted that zeolites sharing the lowest distances
according to our metric have often been synthesized together (see Table S1 in
the Supporting Information). Recovering pairs of structurally similar
frameworks such as ITH-ITR, SBS-SBT, or MWF-PAU at low distance already
suggests that the similarity metric is qualitatively sound. To generalize this
observation, we charted a map of zeolite structures based on their distances.
Figure 2 shows the minimum spanning tree created by converting the AMD
distance matrix into a graph with weighted edges. Although the tree shows
discrete connections and may not be accurate in the presence of outliers, it
facilitates a qualitative interpretation of the results and may provide
insights about synthesis-structure maps. Even without considering synthesis
labels of the data in Fig. 2, known relationships between zeolites emerge
naturally from the structural tree map. Zeolites known for their similar
building patterns are clustered together in the minimum spanning tree,
demonstrating that their AMD values capture the space of zeolites without
learnable features. Examples of such clusters include the ABC-6 zeolites,
structures containing lov building units, six-membered rings frameworks (e.g.,
GIU cluster), Ge- or boron-containing zeolites (e.g., BEC or IRR and SFN-SSF
branch, respectively), to name a few (see also Fig. S3 for a visual guide).
Similarly, structural outliers such as the low-density RWY or JSR, or
interrupted frameworks such as -CLO, -SYT, and -ITV tend to cluster together,
as their distances to all other zeolites is high (Fig. S2). At the same time,
other interrupted frameworks produced in different synthesis conditions, such
as -IRY, -IFU, or -IFT synthesized with germanium, appear along with other
germanosilicates in the synthesis map, not only with other interrupted
frameworks. This ability to traverse the structural space in a continuous way
allows drawing non-obvious connections that may be overlooked by building
units alone. For example, the MEI framework lies within the SBS, SBT, SAO, and
SBE cluster despite not having common CBUs with any of these zeolites.
Nevertheless, the MEI structure also has large pores and building patterns
that could help inform the synthesis of other of these zeolites. In fact, the
recent synthesis of PST-32 (SBT) and PST-2 (SBS/SBT) aluminosilicate zeolites
was performed in the presence of sodium 24, similar to the original inorganic
synthesis of MEI, although at lower Si/Al ratios.

Figure 2: Minimum spanning tree of 253 zeolites in the IZA database according
to the $L_{\infty}$ metric on AMD vectors of length $k=100$ atomic neighbors.
Each framework is a node, and edges minimize the total length of a tree.
Darker (lighter) colors indicate that silicon is more (less) frequent in the
synthesis of each zeolite.

Despite the usefulness of the tree map in connecting zeolites with similar
synthesis conditions, the visual analysis cannot determine whether the map
consistently provides new insights on the synthesis of zeolites. To improve
this qualitative analysis, we performed a hierarchical clustering of the data
to quantify whether the structural distance metric clusters the data according
to the literature synthesis conditions (see Methods). The dendrogram of the
AMD values (Figs. S5 and S6) shows how zeolites are related to each other
based on distances, thus providing a more quantitative view to the minimum
spanning tree of Fig. 2. Then, to create labels for synthesis conditions, we
started with a dataset of extensive synthesis conditions extracted from the
zeolite literature from Jensen et al. 43. After augmenting the data with
frameworks not typically reported in publications, such as those found as
minerals, we analyzed the frequency of occurrence of each synthesis condition
for each framework. Although the initial dataset had information on both
organic and inorganic conditions, we disregarded the organic templates when
labeling the data, thus assuming that inorganic and organic conditions can, to
an extent, be predicted independently of each other. Furthermore, given the
scarcity of data for some synthesis conditions, we focused only on the 14
inorganic conditions that have been used to synthesize at least 10 zeolites,
namely Al, B, Be, Ca, Co, F, Ga, Ge, K, Mg, Na, P, Si, and Zn. Finally, we
verify whether flat clusters formed by points with a maximum distance of each
other share the same positive labels. This intuition is quantified by
computing the homogeneity between data points given clusters formed by a given
distance threshold 44 (see Methods). If all clusters had only positive labels,
their homogeneity would be 1, whereas zero homogeneity indicates perfect
mixing of positive and negative labels. Figure 3a shows that clusters with at
least one positive data point become more homogeneous as the distance
threshold decreases. This supports the qualitative view that structures
considered similar according to the AMD values also share similar synthesis
conditions more often than not. On the other hand, as clusters become larger
and the increasingly dissimilar structures are grouped together, the
homogeneity decreases. Whereas the distribution of labels for some inorganic
agents such as Al, Si, Be, F, or Na exhibit higher homogeneity at low
distances (see Fig. S7), others such as Co, Mg, or Zn show little predictive
power. This suggests that structural distances computed with the AMD have
stronger correlations with certain synthesis conditions than with others.
Although the lack of true negative data points may also influence the computed
homogeneity, these results demonstrate quantitatively that some synthesis-
structure relationships can be established in zeolites using AMDs.

Figure 3: Unsupervised learning for inorganic synthesis of known zeolites. a,
Cluster homogeneity of zeolites for selected elements (see also Fig. S7). b,
Minimum spanning tree of zeolites (Fig. 2) labeled according to frequency of
Ge or Na in the synthesis of each zeolite (see also Fig. S4). Darker (lighter)
colors indicate that the inorganic synthesis condition is more (less) frequent
in the synthesis of each zeolite. c, Subset of the zeolite dendrogram for
selected regions of the minimum spanning tree.

Additional investigations of the data explain the patterns in homogeneity
obtained above. Figure 3b shows how the minimum spanning tree can be
visualized according to the frequency of certain inorganic conditions in
zeolite synthesis (see Fig. S4 for complete results). For example, some
frameworks realizable with Ge or Na form their own groups in the tree, as also
illustrated by the subclusters in dendrograms (Fig. 3c). Indeed, zeolites such
as BEC, ISV, ITT or IWR are typical examples of large- and extra-large pore
structures synthesized using germanium. Similarly, denser phases such as GIU,
FAR, LOS etc. are often obtained in sodium-mediated syntheses. For common
synthesis conditions such as silicon, trends can be derived from the
visualization of silicon-free routes. The labeled trees from Figs. 2 and S4
show that non-silica zeolites are often located in similar regions of the
structure space. Groups formed by zeolites such as NAT, EDI, and THO, or AFO,
AEL, AHT show that non-silica zeolites also share structural patterns that may
be harder to obtain in silica-based structures.

This unsupervised analysis demonstrates that our distance metric relates
zeolites with similar inorganic synthesis conditions without any learnable
parameters. Although zeolite structures contain several outliers and lack true
negative data, the structural patterns still provide a strong prior for
exploring the synthesis conditions. In particular, as inorganic synthesis
conditions can be inferred by the similarity between crystal structures, they
can also help downselect structures for zeolites yet to be realized.

###  2.2 Interpretable classifiers for predicting inorganic synthesis
conditions

One disadvantage of the pure unsupervised learning approach is the inefficient
utilization of the available labels. Although similarity between crystal
structures is a good indicator of common synthesis conditions, the
dissimilarity between structures can also provide insights on which structures
are less likely to be synthesized with a given composition. To perform this
analysis, we use the labeled data to train supervised learning methods that
predict the synthesis conditions of a zeolite given its distances to known
frameworks. Specifically, we trained logistic regression, random forest, and
XGBoost classifiers on literature data to predict each class label
individually. However, training models on the literature labels has two
caveats: (1) the data is often unbalanced, i.e., the number of positive data
points is much smaller than the number of negative data points; and (2) the
negative data is not truly negative, as its lack of literature reporting does
not imply that a zeolite cannot be synthesized under the synthesis conditions
in analysis. To account for these problems, we trained balanced classifiers by
subsampling the dataset for each synthesis conditions, thus ensuring that
training sets had the same proportion of positive and negative data points,
but validation/test sets were allowed to have more negative samples than
positive ones. In that case, because models were tested on different negative
splits, they were prevented from memorizing “negative” data points as truly
negative, as exemplified by the case of MEI zeolite above. Finally, for each
synthesis condition, we performed an extensive hyperparameter optimization for
each of the three classifiers, evaluating the models according to their
accuracy, precision, recall, F1 score, and areas under the receiving operating
characteristic (ROC) and precision-recall (PR) curves.

The results of the hyperparameter search are summarized in Fig. S8. Whereas no
classifier outperforms the other in all tasks, XGBoost models often show the
best metrics of ROC and PR areas under the curve (AUC) for a variety of
synthesis conditions. When the results for the XGBoost classifier are
visualized according to all metrics at once (Fig. 4a), they demonstrate how
the best hyperparameters lead to adequate figures of merit based on results
from the validation set (see also Fig. S9). When evaluated against a held-out
test set, the model with best set of hyperparameters still exhibits high ROC
and PR AUCs for a variety of synthesis conditions (Fig. S10). Nevertheless,
this set of hyperparameters is far from being the only one that performs well
in these conditions (Fig. S11). As discussed in the analysis using
unsupervised learning, the ability to correctly label zeolites whose synthesis
contains Co or Zn is smaller than other labels, as indicated by the worse
performance of all classifiers in labeling these conditions. However, some
synthesis conditions that were not well-predicted by the unsupervised learning
method, such as Mg, can now be predicted using XGBoost models, despite its low
recall (Fig. S10). These results show that ML classifiers can predict
inorganic synthesis conditions using distances towards known zeolites as
features. This has useful implications, as it bypasses the need to create
general representations for zeolites, and instead uses a set of points in the
space as references for new synthesis conditions.

Figure 4: Supervised learning of inorganic synthesis in known zeolites. a,
Results of hyperparameter optimization of XGBoost classifiers for selected
inorganic conditions. Each thin gray line is one set of parameters for the
XGBoost models. The pink line represents the best performance according to the
ROC and PR AUC. The figures of merit are: accuracy (Acc), precision (Prec),
recall (Rec), F1-score (F1), receiving operating characteristic AUC (ROC), and
precision-recall AUC (PR). The figure of merit has value 0 at the center and 1
at the border of the circle. b, Pearson correlation coefficient between AMD
and SHAP values. A negative correlation (green) indicates that smaller
distances lead to higher SHAP values, and thus contribute to classifying the
zeolite with a positive label.

To interpret the outcomes of the classifiers, explainability techniques can be
used to probe what features most affect a positive (or negative)
classification of a zeolite within certain synthesis conditions. Given that
the input features are distances towards known frameworks, a trained
classifier decides how to assign a label to an input structure based on its
similarity values. Using the Shapley value method (SHAP), we analyze what
distances most affect the classification of a zeolite into a given class. As
each SHAP value indicates how much each feature affects the probability of
classifying a framework into a given class, we compute the Pearson correlation
coefficient between the actual feature value and the SHAP value for each one
of the inorganic synthesis conditions. This quantifies whether being close to
a particular framework (feature) increases or decreases the likelihood of
being assigned a positive label. The results for the interpretability of
XGBoost classifiers are shown in Fig. 4b (see also Fig. S12 and S13). As the
correlation coefficient between AMD and SHAP values are computed on a per-
feature (thus per-zeolite) basis, the nodes from the tree map in Fig. 2 are
colored according to these coefficients. In this plot, a negative correlation
(in green color) indicates that low distances increase the SHAP value and thus
the likelihood of being classified as a positive label. Conversely, a positive
correlation (in pink color) with a feature indicates that a given zeolite is
more likely to be synthesized with a given synthesis condition if it is
distant from the featurizing structure. The results not only support the
observations highlighted by the unsupervised learning methods, but also
complement them with new insights. For instance, zeolites synthesized with Ca
and K have a wide overlap of positive and negative correlations (Fig. S13),
possibly due to the clustering of minerals in the tree map. There is also an
overlap between boron-containing zeolites and germanium-containing zeolites,
especially in the ISV branch. This result could be interesting if validated in
practice, especially as the use of boron may enable the removal of Ge from the
synthesis of certain zeolites, such as BEC. The central branch characterized
by Ge-containing zeolites (such as BEC, ISV, IRR, ITT, see Fig. 2) also have
features that correlate with F or Mg, but not Al or Ca. On the other hand, Be-
containing zeolites are often complementary to Si-containing zeolites, as the
former are found only in specific clusters or outliers in the tree map. This
further supports the fact that the classifiers are able to obtain correlations
beyond existing heuristics, thus providing data-driven ways to guide inorganic
synthesis in zeolites.

###  2.3 Proposing inorganic synthesis conditions for hypothetical zeolites

Given that structural similarity is correlated to inorganic synthesis in
zeolites and that supervised learning methods are able to predict synthesis
using only distances as inputs, we can use the models developed in this work
to propose inorganic synthesis conditions for hypothetical zeolites. This
approach complements previous work on the design of OSDAs for frameworks, thus
enabling inorganic synthesis conditions to be predicted in silico whenever a
new framework is proposed. To do that, we used the dataset of 331,171
hypothetical zeolites proposed by Pophale et al. 33, known as the “Deem
dataset.” As structural features and densities from the hypothetical zeolites
optimized with force fields may deviate from the experimental ones, we used
the IZA and hypothetical zeolites from Erlebach et al. 45, which employed a
neural network force field trained at the SCAN level of density functional
theory calculations to reoptimize the hypothetical zeolites. Then, by
comparing the hypothetical structures against all known zeolites, we created a
distance matrix that is used as input for the unsupervised and supervised
learning methods shown in the previous section. As in the case of known
zeolites, AMDs are correlated with differences of density, but are not solely
determined by them (Fig. S14). Using AMDs, a low-dimensional map can be
created for all hypothetical structures, thus providing an intuitive way to
visualize the space of structures. Figure S15 shows a 2D projection of the
distribution of hypothetical zeolites based on their distance matrix using
UMAP. This plot shows that distance features are able to sort the space of
zeolites according to energy and density despite not using this information as
explicit inputs. The visualization also illustrates that most hypothetical
frameworks do not have neighboring known structures. While 105 of all known
zeolites have at least one other known zeolite up to 0.1 Å away (45% of
structures, see Fig. S6), only about 36,112 of the 331,171 hypothetical
structures have at least one known zeolite as neighbor when the same distance
threshold is used (11% of zeolites in the dataset). This illustrates how the
space of enumerated zeolites is often populated with structures far from known
structural patterns of zeolites, as also demonstrated by previous studies (see
also Fig. S16).

As demonstrated in this work, zeolites in the neighborhood of known frameworks
are likely to share similar synthesis conditions as those known structures.
Thus, downselecting frameworks for given synthesis conditions can benefit from
the unsupervised and supervised methods developed here. This approach can be
used in combination with previous “synthesizability metrics” of zeolites, such
as local interatomic distances 35 or other data-driven predictions 46.
However, we chose to evaluate them independently, as these synthesizability
predictions do not take into account that certain known frameworks may be
considered “unfeasible” depending on the synthesis conditions 47, 34. For
instance, structures containing three-connected rings, such as those with
building units lov or vsv, could be ranked as “unsynthesizable,” despite being
achieved with beryllium or borogermanate conditions. Thus, to propose
synthesis conditions for zeolites, we evaluated all hypothetical frameworks
for all synthesis conditions using an ensemble of 100 binary classifiers per
inorganic condition (see Methods). As each classifier is trained on different
negative data splits, the resulting classification varies for each model,
allowing us to assess the degree of agreement between the models. By taking
the average of the predictions, we obtain the probability of synthesizing the
zeolite with that synthesis condition.

Figure 5: Selection of hypothetical zeolites using AMD values and inorganic
conditions. a, Selection of LTA-like zeolites according to predicted Si-based
recipes. Only the 1000 closest points to LTA are shown in this graph. b,
Unusual cages that distinguish the two hypothetical structures #308,105 and
#313,030 and determine their synthesis to be more/less likely to be successful
under Si conditions, respectively. c, Three examples of hypothetical zeolites
selected based on the predicted synthesis conditions. Only five (Al, Be, Ge,
K, Si) of the 14 synthesis conditions are shown.

Figure 5a depicts the distribution of hypothetical zeolites with Si-based
recipes in the neighborhood of LTA zeolite. As all distances between known and
hypothetical zeolites had been already computed, we can use both the distances
and the class probabilities as criteria for navigating the space of
hypothetical structures. This navigation using reference materials instead of
features facilitates the selection process and can also inform their
synthesis. For example, Figs. 5a,b illustrate two different hypothetical
zeolites in the neighborhood of LTA. Although both have low distance towards
LTA (compare with dendrogram in Fig. S6), structure #308,105 is predicted to
be more likely to be synthesized as a silicate than #313,030. Both contain the
lta and sod cages characteristic of the LTA zeolite, but differ by the
presence of a second cage similar to sod, shown in Fig. 5b. Whereas this new
building unit resembles an expanded sod cage with distorted six-membered rings
in #308,105, hypothetical framework #313,030 shows a new cage, formed by the
merging of two sod cages, not seen in known zeolites. This increased distance
towards known structural patterns drives the prediction of feasible synthesis
using Si as unlikely, even when the distance towards the LTA zeolite is lower.
This example shows how the combination of AMD values and classifier
predictions facilitates the exploration of hypothetical zeolites using
reference structures.

Beyond exploration of the zeolite space, the models also uncover existing and
new synthesis-structure relationships. Figure 5c shows three examples of
hypothetical frameworks predicted to be synthesized using three different
elements: Be, Ge, and K. To obtain these frameworks, we filtered only
frameworks within densities of 14 and 17 T/1000 Å3 that are predicted to have
100% probability of synthesis with the given element. Then, we ranked the
frameworks by their relative energy. Despite not using explicit labels on the
CBUs, the supervised learning models recovered the known heuristics of
building units and inorganic synthesis conditions. For instance, framework
#261,338, predicted to be synthesized in presence of Be, is formed mostly by
lov building units, as found in other Be-zeolites such as RSN, LOV, or NAB.
This same framework is predicted to be unlikely as a silicate, possibly
following the trends seen in JSR or NPT structures. Hypothetical zeolite
#64,550, predicted to be synthesized with germanium, also shows features
similar to known ones. In addition to its three-dimensional pore structure,
with $12\times 12\times 10$ intersecting pores, the structure shows the d4r
CBU typical of other structurally similar germanosilicates, such as POS or
UOV, but with 7 symmetrically inequivalent T sites. Finally, one unrealized
framework predicted to be synthesized with potassium is structure #303,768.
Although this hypothetical structure does not exhibit typical CBUs, the local
structures similar to d8r CBUs are predicted to be favored by K, in analogy
with similar relationships in known zeolites. This demonstrates how data-
driven models can not only recover known relationships between CBUs and
inorganic conditions, but also propose new synthesis-structure relationships
in zeolites based on distance patterns towards known structures. When used to
analyze the entire space of hypothetical frameworks, the models show that the
distribution of predicted inorganic synthesis conditions is uneven across the
space of zeolites (Fig. S19). For instance, whereas about 27% of all known
zeolites can be synthesized with germanium, according to the literature
dataset we used in this work, only 8% of the hypothetical zeolites have
probability of being synthesizable under Ge conditions of at least 80%.
Similarly, the space of hypothetical structures is surprisingly lacking in
structures predicted to be synthesizable with sodium. While about 45% of all
known structures have at least one sodium-based synthesis, 17% of hypothetical
structures are predicted to be realizable with Na given the 80% threshold
probability. As most enumerated datasets are often created without considering
synthesis conditions 48, comprehensive enumerations may introduce biases in
structures that do not reflect the space of zeolite synthesis typically
considered in practice. Thus, in combination with template design 17 and
property screening 49, 50, our methods to predict inorganic synthesis
conditions in zeolites may help in synthesizing unrealized frameworks with
targeted properties.

##  3 Conclusions

Mapping the space of inorganic conditions in materials synthesis is an
outstanding challenge due to the complexity of chemical interactions during
synthesis. In the case of zeolites, synthesis conditions are known to affect
structural patterns in the materials, but finding correlations between
structural patterns and inorganic syntheses often relies on heuristics. In
this work, we used unsupervised and supervised learning methods to propose
inorganic synthesis conditions for zeolite synthesis. In particular, we showed
how a mathematically strong distance metric between crystals can predict
inorganic synthesis conditions in zeolites. This enables structural
comparisons beyond human-crafted labels of building units or pore
sizes/topologies. Clustering techniques demonstrate that our metric
consistently recalls the inorganic synthesis conditions from literature
datasets, thus providing predictive power even in the absence of labels. Then,
we show that ML classifiers can predict synthesis conditions of a given
framework based on its distribution of distances towards known structures. The
classifiers were used to predict 14 different synthesis conditions for known
and unrealized zeolites. When explaining the predictions, we showed how ML
classifiers analyze synthesis conditions also from the dissimilarity between
crystals, as well as from the similarity. The results from the explainability
analysis reveals overlaps in inorganic synthesis conditions, such as boron and
germanium, as well as complementary relationships, such as silicon and
beryllium. Finally, we showcased how our methods can be used to predict
inorganic synthesis conditions for hypothetical zeolites, facilitating the
downselection of new structures for experimental attempts. This combination of
data-driven methods can create a pathway for full in silico prediction of
zeolite synthesis beyond the design of organic templates.

##  4 Methods

###  4.1 Pointwise Distance Distributions and Average Minimum Distances

Any periodic crystal structure is modeled as a periodic set $S$ of atomic
centers considered as zero-sized points, with atomic types as optional labels.
Any linear basis of vectors $v_{1},v_{2},v_{3}$ in 3-dimensional space
generates a lattice $\Lambda=\\{c_{1}v_{1}+c_{2}v_{2}+c_{3}v_{3}\mid
c_{i}\text{ are integers}\\}$ and unit cell
$U=\\{t_{1}v_{1}+t_{2}v_{2}+t_{3}v_{3}\mid 0\leq t_{i}<1\\}$. Any finite motif
of points $M\subset U$ defines the periodic point set $S=\\{p+v\mid p\in
M,v\in\Lambda\\}$. This conventional representation of a periodic crystal $S$
by a unit cell and a motif is ambiguous because infinitely many different
pairs (cell, motif) generate periodic sets that are equivalent up to rigid
motion (a composition of translations and rotations). Fixing any reduced cells
leads to unavoidable discontinuities 51 even for 2-dimensional lattices.

The ambiguity of crystal representations was theoretically resolved for all
periodic point sets in any dimension by the complete isoset 52 invariant. We
define below the computationally faster Pointwise Distance Distribution (PDD)
invariant, which distinguished all (more than 670,000) periodic crystals in
the Cambridge Structural Database (CSD) through more than 200 billion pairwise
comparisons within two days on a typical desktop computer.

Fix a number $k\geq 1$ of atomic neighbors. Our experiments on zeolites and
the CSD used $k=100$. Let $S$ be a periodic set with a motif $M$ of points
$p_{1},\dots,p_{m}$. For each point $p_{i}$, write down the sequence of
increasing distances $d_{i1}\leq\cdots\leq d_{ik}$ to its $k$ nearest
neighbors in the full infinite set $S$ without considering any extended cell
or cut-off radius. Collect these sequences of distances into an $m\times k$
matrix and lexicographically order the rows. If any $l$ of the rows coincide
(usually due to extra symmetries), collapse them into a single row with the
weight $l/m$ and put these weights into an additional first column (unique
rows have weight $1/m$). The resulting $m\times(k+1)$ matrix
$\mathrm{PDD}(S;k)$ is called the _Pointwise Distance Distribution_ , a
statistical distribution of rows with weights describing each point’s
environment. As an example, Fig. 6 shows the computation for a point in the
square lattice $S$ whose first $k=8$ neighbours have distances $1,1,1,1$ (in
green) and $\sqrt{2},\sqrt{2},\sqrt{2},\sqrt{2}$ (in blue).

Figure 6: Computational pipeline of PDD is illustrated for a 2-dimensional
square lattice.

The _Average Minimum Distance_ $\mathrm{AMD}(S;k)$ is the vector obtained by
taking the weighted average of the last $k$ columns in $\mathrm{PDD}(S;k)$, so
$\mathrm{AMD}$ is a single vector of $k$ average distances. To compare two AMD
vectors of the same length, our experiments used the $L_{\infty}$ metric equal
to the maximum absolute difference of corresponding coordinates. For a metric
on PDDs, we use the Earth Mover’s Distance (EMD)53 with the $L_{\infty}$
metric on rows. If any point of $S$ is perturbed in its
$\varepsilon$-neighborhood, then $\mathrm{PDD}(S;k)$ changes by at most
$2\varepsilon$ in the EMD metric. If a periodic set $S$ is generic, which is
achievable by almost any perturbation of atoms, then $S$ can be reconstructed
from the number $m$ of motif points, a (basis of a) lattice $\Lambda$ and
$\mathrm{PDD}(S;k)$ with a known upper bound on $k$. For the details on these
results, see Definition C5 and proofs of Theorems 4.3 and 4.4 in the extended
version of Ref. 54.

###  4.2 Zeolite structures data

The dataset of 253 known zeolite structures used in the unsupervised learning
method was obtained from the International Zeolite Association (IZA) database
55. The dataset of hypothetical frameworks used in this work was developed by
Pophale et al. 33, and re-optimized using a neural network force field trained
on DFT-SCAN data by Erlebach et al. 45. Because not all of the 253 known
zeolites used previously were optimized by Erlebach et al., we used their
subset of 236 known frameworks when computing distance matrices from the
hypothetical frameworks and the known frameworks.

Following the notation from the IZA, known zeolites are named in this
manuscript according to their three-letter code in bold typeface. Known CBUs
are represented with their three-letter code in lowercase and italic typeface.

###  4.3 Literature data

Literature data was obtained from public datasets of zeolite synthesis
conditions from Jensen et al. 43, which has been validated by Schwalbe-Koda et
al. for the computational design of OSDAs 17. Whereas in those works only
zeolite-OSDA pairs were considered, in this work only relationships between
zeolites and non-organic synthesis conditions are analyzed. As the dataset was
produced by collecting literature data from over 60 years of studies in
synthetic zeolites, several natural frameworks were omitted from the table, as
well as newer structures not captured at the time of this study. To address
this problem, we manually inserted new rows on the table with the composition
of the missing structures. The compositions were obtained with manual
verification of the synthesis conditions in articles describing the
mineral/synthetic zeolite, as also shown in the online IZA database. The
resulting, cleaned data used in this study is available for download (see Code
and Data Availability).

In the literature analysis, a zeolite is classified as having a certain
synthesis condition when at least 25% of its synthesis recipes exhibit that
condition (excluding OSDAs). This label is used as a categorical variable when
performing the classification task.

###  4.4 Unsupervised learning

A minimum spanning tree between zeolites was constructed by first creating a
fully connected, undirected graph with weighted edges, where weights
correspond to the distances between two structures. The tree was then obtained
using NetworkX’s (v. 2.5) 56 minimum spanning tree algorithm, which minimizes
the total length of the tree.

The dendrogram of known zeolites was produced by creating a linkage matrix
from the distance matrix using the Ward algorithm as implemented in SciPy (v.
1.10.0) 57. The resulting clusters in Fig. S6 were obtained by forming flat
clusters with maximum AMD distance of a given threshold.

The homogeneity of the clustering was computed by calculating the Shannon
entropy of flat clusters created with a given threshold 44, as implemented in
scikit-learn (v. 1.2.0) 58. As the literature dataset is not balanced and lack
true negative points, the homogeneity was only computed for clusters
containing at least one positive data point. This ensures that a large
homogeneity corresponds to recall of positive data points, which prevents
biasing this metric in imbalanced datasets.

Dimensionality reduction was performed using UMAP 59, as implemented in the
umap-learn package in Python (v. 0.5.3). The 2D UMAP plot was produced by
comparing hypothetical frameworks using the cosine distance of their
normalized distances to IZA structures, and using 10 neighbors as parameter.

###  4.5 Supervised learning

Classification of inorganic synthesis conditions was performed by training
separate classifiers for each synthesis condition. The features used during
training were the distances towards the 253 known frameworks, as computed with
the AMD method described above. To obtain a statistically meaningful result,
only elements used to synthesize at least 10 zeolites were considered. In
particular, 14 inorganic conditions are considered: Al, B, Be, Ca, Co, F, Ga,
Ge, K, Mg, Na, P, Si, and Zn.

Train-validation-test sets were created starting with a 60-20-20 ratio,
respectively, then subsampling the training set to have an equal number of
points with positive and negative labels. Although techniques such as
reweighting or resampling could have been employed to obtained balanced
training sets, removing data points is a simple approach that prevents
classifiers from treating negative data as “true negative”, resembling
positive-unlabeled learning strategies.

Hyperparameter optimization of synthesis classifiers was performed using a
grid-search method over relevant spaces of hyperparameters for logistic
regression, random forest, and XGBoost 60 methods. The full range of
hyperparameters investigated in this hyperparameter search is shown in Tables
S2, S3 and S4, following the notation in the scikit-learn (v. 1.2.0) and
xgboost (v. 1.7.5) Python packages. Model performances were compared using the
same dataset splits, and the best model is selected according to its
validation performance. The results on the paper showcase the performance on
held-out test data.

One of the best models to classify synthesis conditions of zeolites was the
XGBoost model with the following hyperparameters: colsample_bytree = 0.5,
learning_rate = 0.1, max_depth = 6, min_child_weight = 1, n_estimators = 200,
subsample = 0.5. This model and set of hyperparameters showed good performance
across a range of inorganic synthesis conditions, as evaluated by the
accuracy, precision, recall, F1 score, area under the receiving operator
characteristic curve (ROC AUC), and area under the precision-recall curve (PR
AUC). In particular, the best model was selected to maximize the ROC AUC and
PR AUC for the balanced classifiers. As a comparison, the performance metrics
and their baselines of unbalanced classifiers — thus trained on dataset splits
with an uneven number of positive/negative labels — are shown in Fig. S10.

Explainability of the classifiers was computed using the Shapley value method
(SHAP) 61 under the TreeExplainer framework 62, as implemented in the shap
Python package (v. 0.41.0). The interventional feature perturbation method was
used without limit for the tree explainer. Then, correlations between the SHAP
values and the distance features were computed for each synthesis condition.
To ensure that the correlations are not artifacts of particular train splits,
we report the average correlation obtained from an ensemble of 100 XGBoost
classifiers trained on splits with different negative data points.

This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory (LLNL) under Contract DE-
AC52-07NA27344. D.S.-K. acknowledges funding from the Laboratory Directed
Research and Development (LDRD) Program at LLNL under project tracking code
22-ERD-055, and under the Postdoctoral Development Program. T.A.P.
acknowledges funding from the Center for Enhanced Nanofluidic Transport
(CENT), an Energy Frontier Research Center funded by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences under Award DE-SC0019112.
D.E.W. and V.A.K. acknowledge funding from the EPSRC (EP/R018472/1,
EP/X018474/1) and Royal Academy of Engineering (IF2122/186) in the UK.

Manuscript released as LLNL-JRNL-851183.

## Data and Code Availability

The code to compute PDDs and AMDs for arbitrary crystal structures is
available on GitHub at https://github.com/dwiddow/average-minimum-distance.
The code and data required to reproduce all results and figures in this
manuscript will be available after peer-review.

## References

  * Davis 2002 Davis, M. E. Ordered porous materials for emerging applications. _Nature_ 2002, _417_ , 813–821
  * Čejka et al. 2007 Čejka, J., van Bekkum, H., Corma, A., Schueth, F., Eds. _Introduction to Zeolite Science and Practice_ , 3rd ed.; Studies in Surface Science and Catalysis 168; Elsevier: Amsterdam ; Boston, 2007; OCLC: ocn163318390
  * Vermeiren and Gilson 2009 Vermeiren, W.; Gilson, J.-P. Impact of Zeolites on the Petroleum and Petrochemical Industry. _Topics in Catalysis_ 2009, _52_ , 1131–1161
  * Li et al. 2017 Li, Y.; Li, L.; Yu, J. Applications of Zeolites in Sustainable Chemistry. _Chem_ 2017, _3_ , 928–949
  * Li and Yu 2021 Li, Y.; Yu, J. Emerging Applications of Zeolites in Catalysis, Separation and Host–Guest Assembly. _Nature Reviews Materials_ 2021,
  * Dusselier and Davis 2018 Dusselier, M.; Davis, M. E. Small-Pore Zeolites: Synthesis and Catalysis. _Chemical Reviews_ 2018, _118_ , 5265–5329
  * Corma 2004 Corma, A. In _Studies in Surface Science and Catalysis_ ; van Steen, E., Claeys, I., Callanan, L., Eds.; Elsevier B.V., 2004; Vol. 154 A; pp 25–40
  * Cundy and Cox 2003 Cundy, C. S.; Cox, P. A. The Hydrothermal Synthesis of Zeolites: History and Development from the Earliest Days to the Present Time. _Chemical Reviews_ 2003, _103_ , 663–701
  * Lobo et al. 1995 Lobo, R. F.; Zones, S. I.; Davis, M. E. Structure-Direction in Zeolite Synthesis. _Journal of Inclusion Phenomena and Molecular Recognition in Chemistry_ 1995, _21_ , 47–78
  * Barrer 1981 Barrer, R. M. Zeolites and Their Synthesis. _Zeolites_ 1981, _1_ , 130–140
  * Brand et al. 2017 Brand, S. K.; Schmidt, J. E.; Deem, M. W.; Daeyaert, F.; Ma, Y.; Terasaki, O.; Orazov, M.; Davis, M. E. Enantiomerically Enriched, Polycrystalline Molecular Sieves. _Proceedings of the National Academy of Sciences of the United States of America_ 2017, _114_ , 5101–5106
  * Gallego et al. 2017 Gallego, E. M.; Portilla, M. T.; Paris, C.; León-Escamilla, A.; Boronat, M.; Moliner, M.; Corma, A. ”Ab Initio” Synthesis of Zeolites for Preestablished Catalytic Reactions. _Science_ 2017, _355_ , 1051–1054
  * Lewis et al. 1996 Lewis, D. W.; Willock, D. J.; Catlow, C. R. A.; Thomas, J. M.; Hutchings, G. J. De novo design of structure-directing agents for the synthesis of microporous solids. _Nature_ 1996, _382_ , 604–606
  * Sastre et al. 2005 Sastre, G.; Cantin, A.; Diaz-Cabañas, M. J.; Corma, A. Searching Organic Structure Directing Agents for the Synthesis of Specific Zeolitic Structures: An Experimentally Tested Computational Study. _Chemistry of Materials_ 2005, _17_ , 545–552
  * Schmidt et al. 2015 Schmidt, J. E.; Deem, M. W.; Lew, C.; Davis, T. M. Computationally-Guided Synthesis of the 8-Ring Zeolite AEI. _Topics in Catalysis_ 2015, _58_ , 410–415
  * Davis et al. 2016 Davis, T. M.; Liu, A. T.; Lew, C. M.; Xie, D.; Benin, A. I.; Elomari, S.; Zones, S. I.; Deem, M. W. Computationally Guided Synthesis of SSZ-52: A Zeolite for Engine Exhaust Clean-Up. _Chemistry of Materials_ 2016, _28_ , 708–711
  * Schwalbe-Koda et al. 2021 Schwalbe-Koda, D.; Kwon, S.; Paris, C.; Bello-Jurado, E.; Jensen, Z.; Olivetti, E.; Willhammar, T.; Corma, A.; Román-Leshkov, Y.; Moliner, M.; Gómez-Bombarelli, R. A Priori Control of Zeolite Phase Competition and Intergrowth with High-Throughput Simulations. _Science_ 2021, _374_ , 308–315
  * Schwalbe-Koda et al. 2021 Schwalbe-Koda, D.; Corma, A.; Román-Leshkov, Y.; Moliner, M.; Gómez-Bombarelli, R. Data-driven design of biselective templates for intergrowth zeolites. _The Journal of Physical Chemistry Letters_ 2021, _12_ , 10689–10694
  * Bello-Jurado et al. 2022 Bello-Jurado, E.; Schwalbe-Koda, D.; Nero, M.; Paris, C.; Uusimäki, T.; Román-Leshkov, Y.; Corma, A.; Willhammar, T.; Gómez-Bombarelli, R.; Moliner, M. Tunable CHA/AEI zeolite intergrowths with A priori biselective organic structure-directing agents: controlling enrichment and implications for selective catalytic reduction of NOx. _Angewandte Chemie International Edition_ 2022, _61_ , e202201837
  * Di Iorio et al. 2020 Di Iorio, J. R.; Li, S.; Jones, C. B.; Nimlos, C. T.; Wang, Y.; Kunkes, E.; Vattipalli, V.; Prasad, S.; Moini, A.; Schneider, W. F.; Gounder, R. Cooperative and Competitive Occlusion of Organic and Inorganic Structure-Directing Agents within Chabazite Zeolites Influences Their Aluminum Arrangement. _Journal of the American Chemical Society_ 2020, _142_ , 4807–4819
  * Shin et al. 2019 Shin, J.; Jo, D.; Hong, S. B. Rediscovery of the Importance of Inorganic Synthesis Parameters in the Search for New Zeolites. _Acc. Chem. Res._ 2019, _52_ , 1419–1427
  * Li et al. 2023 Li, X.; Han, H.; Evangelou, N.; Wichrowski, N. J.; Lu, P.; Xu, W.; Hwang, S.-J.; Zhao, W.; Song, C.; Guo, X., et al. Machine learning-assisted crystal engineering of a zeolite. _Nature Communications_ 2023, _14_ , 3152
  * Li et al. 2019 Li, S.; Li, J.; Dong, M.; Fan, S.; Zhao, T.; Wang, J.; Fan, W. Strategies to control zeolite particle morphology. _Chemical Society Reviews_ 2019, _48_ , 885–907
  * Lee et al. 2021 Lee, H.; Shin, J.; Lee, K.; Choi, H. J.; Mayoral, A.; Kang, N. Y.; Hong, S. B. Synthesis of Thermally Stable SBT and SBS/SBT Intergrowth Zeolites. _Science_ 2021, _373_ , 104 LP – 107
  * Oleksiak and Rimer 2014 Oleksiak, M. D.; Rimer, J. D. Synthesis of Zeolites in the Absence of Organic Structure-Directing Agents: Factors Governing Crystal Selection and Polymorphism. _Reviews in Chemical Engineering_ 2014, _30_ , 1–49
  * Jensen et al. 2019 Jensen, Z.; Kim, E.; Kwon, S.; Gani, T. Z. H.; Román-Leshkov, Y.; Moliner, M.; Corma, A.; Olivetti, E. A Machine Learning Approach to Zeolite Synthesis Enabled by Automatic Literature Data Extraction. _ACS Central Science_ 2019, _5_ , 892–899
  * Muraoka et al. 2019 Muraoka, K.; Sada, Y.; Miyazaki, D.; Chaikittisilp, W.; Okubo, T. Linking Synthesis and Structure Descriptors from a Large Collection of Synthetic Records of Zeolite Materials. _Nature Communications_ 2019, _10_ , 4459
  * Asselman et al. 2022 Asselman, K.; Vandenabeele, D.; Pellens, N.; Doppelhammer, N.; Kirschhock, C. E.; Breynaert, E. Structural Aspects Affecting Phase Selection in Inorganic Zeolite Synthesis. _Chemistry of Materials_ 2022, _34_ , 11081–11092
  * Itabashi et al. 2012 Itabashi, K.; Kamimura, Y.; Iyoki, K.; Shimojima, A.; Okubo, T. A Working Hypothesis for Broadening Framework Types of Zeolites in Seed-Assisted Synthesis without Organic Structure-Directing Agent. _Journal of the American Chemical Society_ 2012, _134_ , 11542–11549
  * Li et al. 2015 Li, J.; Corma, A.; Yu, J. Synthesis of New Zeolite Structures. _Chemical Society Reviews_ 2015, _44_ , 7112–7127
  * Blatov et al. 2013 Blatov, V. A.; Ilyushin, G. D.; Proserpio, D. M. The Zeolite Conundrum: Why Are There so Many Hypothetical Zeolites and so Few Observed? A Possible Answer from the Zeolite-Type Frameworks Perceived as Packings of Tiles. _Chemistry of Materials_ 2013, _25_ , 412–424
  * Kuznetsova et al. 2018 Kuznetsova, E. D.; Blatova, O. A.; Blatov, V. A. Predicting New Zeolites: A Combination of Thermodynamic and Kinetic Factors. _Chemistry of Materials_ 2018, _30_ , 2829–2837
  * Pophale et al. 2011 Pophale, R.; Cheeseman, P. A.; Deem, M. W. A Database of New Zeolite-like Materials. _Physical Chemistry Chemical Physics_ 2011, _13_ , 12407–12412
  * Li et al. 2019 Li, L.; Slater, B.; Yan, Y.; Wang, C.; Li, Y.; Yu, J. Necessity of Heteroatoms for Realizing Hypothetical Aluminophosphate Zeolites: A High-Throughput Computational Approach. _The Journal of Physical Chemistry Letters_ 2019, _10_ , 1411–1415
  * Li et al. 2013 Li, Y.; Yu, J.; Xu, R. Criteria for Zeolite Frameworks Realizable for Target Synthesis. _Angewandte Chemie International Edition_ 2013, _52_ , 1673–1677
  * Helfrecht et al. 2019 Helfrecht, B. A.; Semino, R.; Pireddu, G.; Auerbach, S. M.; Ceriotti, M. A New Kind of Atlas of Zeolite Building Blocks. _The Journal of Chemical Physics_ 2019, _151_ , 154112
  * Widdowson et al. 2022 Widdowson, D.; Mosca, M.; Pulido, A.; Cooper, A.; Kurlin, V. Average Minimum Distances of periodic point sets - fundamental invariants for mapping all periodic crystals. _MATCH Communications in Mathematical and in Computer Chemistry_ 2022, _87_ , 529–559
  * Widdowson and Kurlin 2022 Widdowson, D.; Kurlin, V. Resolving the data ambiguity for periodic crystals. _Advances in Neural Information Processing Systems (NeurIPS 2022)_ 2022, _35_ , 24625–24638
  * Elkin and Kurlin 2023 Elkin, Y.; Kurlin, V. A new near-linear time algorithm for k-nearest neighbor search using a compressed cover tree. International Conference on Machine Learning (ICML). 2023
  * Bartók et al. 2013 Bartók, A. P.; Kondor, R.; Csányi, G. On Representing Chemical Environments. _Physical Review B_ 2013, _87_ , 184115
  * Pozdnyakov et al. 2020 Pozdnyakov, S. N.; Willatt, M. J.; Bartók, A. P.; Ortner, C.; Csányi, G.; Ceriotti, M. Incompleteness of atomic structure representations. _Physical Review Letters_ 2020, _125_ , 166001
  * Schwalbe-Koda et al. 2019 Schwalbe-Koda, D.; Jensen, Z.; Olivetti, E.; Gómez-Bombarelli, R. Graph Similarity Drives Zeolite Diffusionless Transformations and Intergrowth. _Nature Materials_ 2019, _18_ , 1177–1181
  * Jensen et al. 2021 Jensen, Z.; Kwon, S.; Schwalbe-Koda, D.; Paris, C.; Gómez-Bombarelli, R.; Román-Leshkov, Y.; Corma, A.; Moliner, M.; Olivetti, E. A. Discovering Relationships between OSDAs and Zeolites through Data Mining and Generative Neural Networks. _ACS Central Science_ 2021, _7_ , 858–867
  * Rosenberg and Hirschberg 2007 Rosenberg, A.; Hirschberg, J. V-Measure: A Conditional Entropy-Based External Cluster Evaluation Measure. Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning (EMNLP-CoNLL). Prague, Czech Republic, 2007; pp 410–420
  * Erlebach et al. 2022 Erlebach, A.; Nachtigall, P.; Grajciar, L. Accurate large-scale simulations of siliceous zeolites by neural network potentials. _npj Computational Materials_ 2022, _8_ , 174
  * Helfrecht et al. 2022 Helfrecht, B. A.; Pireddu, G.; Semino, R.; Auerbach, S. M.; Ceriotti, M. Ranking the Synthesizability of Hypothetical Zeolites with the Sorting Hat. _Digital Discovery_ 2022, _1_ , 779–789
  * Mazur et al. 2016 Mazur, M.; Wheatley, P. S.; Navarro, M.; Roth, W. J.; Položij, M.; Mayoral, A.; Eliášová, P.; Nachtigall, P.; Čejka, J.; Morris, R. E. Synthesis of ’unfeasible’ Zeolites. _Nature Chemistry_ 2016, _8_ , 58–62
  * Argente et al. 2023 Argente, E.; Valero, S.; Misturini, A.; Treacy, M. M.; Baumes, L.; Sastre, G. Computer Generation of Hypothetical Zeolites. _AI-Guided Design and Property Prediction for Zeolites and Nanoporous Materials_ 2023, 145–172
  * Lin et al. 2012 Lin, L.-C.; Berger, A. H.; Martin, R. L.; Kim, J.; Swisher, J. A.; Jariwala, K.; Rycroft, C. H.; Bhown, A. S.; Deem, M. W.; Haranczyk, M.; Smit, B. In Silico Screening of Carbon-Capture Materials. _Nature Materials_ 2012, _11_ , 633–641
  * Hewitt et al. 2022 Hewitt, D.; Pope, T.; Sarwar, M.; Turrina, A.; Slater, B. Machine learning accelerated high-throughput screening of zeolites for the selective adsorption of xylene isomers. _Chemical Science_ 2022, _13_ , 13178–13186
  * Kurlin 2022 Kurlin, V. A. Mathematics of 2-dimensional lattices. _Foundations of Computational Mathematics_ 2022, 1–59
  * Anosova and Kurlin 2021 Anosova, O.; Kurlin, V. An isometry classification of periodic point sets. Lecture Notes in Computer Science (Proceedings of DGMM). 2021; pp 229–241
  * Hargreaves et al. 2020 Hargreaves, C. J.; Dyer, M. S.; Gaultois, M. W.; Kurlin, V. A.; Rosseinsky, M. J. The Earth Mover’s Distance as a Metric for the Space of Inorganic Compositions. _Chemistry of Materials_ 2020, _32_ , 10610–10620
  * Widdowson and Kurlin 2021 Widdowson, D.; Kurlin, V. Pointwise Distance Distributions of periodic sets. _arXiv:2108.04798_ 2021,
  * Baerlocher and McCusker 2023 Baerlocher, C.; McCusker, Database of Zeolite Structures. https://www.iza-structure.org/databases/, 2023
  * Hagberg et al. 2008 Hagberg, A. A.; Schult, D. A.; Swart, P. J. Exploring Network Structure, Dynamics, and Function using NetworkX. Proceedings of the 7th Python in Science Conference. Pasadena, CA USA, 2008; pp 11 – 15
  * Virtanen et al. 2020 Virtanen, P. et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. _Nature Methods_ 2020, _17_ , 261–272
  * Pedregosa et al. 2011 Pedregosa, F. et al. Scikit-learn: Machine Learning in Python. _Journal of Machine Learning Research_ 2011, _12_ , 2825–2830
  * McInnes et al. 2018 McInnes, L.; Healy, J.; Melville, J. UMAP: Uniform manifold approximation and projection for dimension reduction. _arXiv:1802.03426_ 2018,
  * Chen and Guestrin 2016 Chen, T.; Guestrin, C. XGBoost: A Scalable Tree Boosting System. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York, NY, USA, 2016; pp 785–794
  * Lundberg and Lee 2017 Lundberg, S. M.; Lee, S.-I. In _Advances in Neural Information Processing Systems 30_ ; Guyon, I., Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R., Eds.; Curran Associates, Inc., 2017; pp 4765–4774
  * Lundberg et al. 2020 Lundberg, S. M.; Erion, G.; Chen, H.; DeGrave, A.; Prutkin, J. M.; Nair, B.; Katz, R.; Himmelfarb, J.; Bansal, N.; Lee, S.-I. From local explanations to global understanding with explainable AI for trees. _Nature Machine Intelligence_ 2020, _2_ , 2522–5839

Supporting Information for: Inorganic synthesis-structure maps in zeolites
with machine learning and crystallographic distances
Daniel Schwalbe-Koda∗,†, Daniel E. Widdowson,‡ Tuan Anh Pham,† and Vitaliy
Kurlin∗,‡

†Lawrence Livermore National Laboratory, Livermore, CA, United States
‡University of Liverpool, Liverpool, United Kingdom
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  Appendix A Supporting Figures

Figure S1: Correlations of AMD values between zeolites with experimental
lattice parameters from the IZA database and a, density differences, b, AMD
values for zeolites optimized with NNPscan, c, SOAP distance, and d, the graph
distance D-measure. The values from c, d were retrieved from Schwalbe-Koda et
al. (Ref. 42 from the main text) Figure S2: Distributions of AMD distances for
each zeolite against all other known zeolites. Brighter colors indicate a
higher median. All subfigures share the same x axis. Figure S3: Minimum
spanning tree of known zeolites with colored labels for a visual guide.
Although several clusters could be highlighted, only some of those discussed
in the main text are shown here. Figure S4: Minimum spanning tree of known
zeolites labeled according to the fraction of recipes per element. Figure S5:
Complete dendrogram of zeolites created using the AMD values as features.
Figure S6: Subclusters of the full dendrogram in Fig. S5 with labels. Figure
S7: Clustering homogeneity for different synthesis conditions when flat
clusters are created with the given threshold. A homogeneity of zero indicates
a perfect mixing between positive and negative labels in the same clusters.
Figure S8: Precision-recall (PR) and Receiver Operating Characteristic (ROC)
areas under the curve (AUC) for different hyperparameters of three different
classifiers: logistic regression, random forest, and XGBoost. The PR AUC and
ROC AUC are adopted as the main figures of merit for evaluating these
classifiers. Figure S9: All figures of merit for different hyperparameters of
the XGBoost classifiers. Each gray line represents the average metric of five
runs at each set of hyperparameters. The figures of merit of interest, along
with the values of each line in the circle, are shown in the lower right of
the plot. Figure S10: Precision (PR) and recall (RC) curves for the best a,
balanced and b, unbalanced XGBoost classifier. The receiver operating
characteristic curve, shown with the true positive (TP) and false positive
(FP) ratios, are also depicted for a c, balanced and d, unbalanced XGBoost
classifier. The best hyperparameters are selected according to the validation
results. The curves in this figure are computed for a held-out test split.
Figure S11: Mean area under the ROC and PR curves for the XGBoost classifiers,
computed with respect to all synthesis predictions at once. The best models
maximize both the PR and ROC curves. Figure S12: Visualization of the SHAP
values for a single XGBoost classifier predicting whether Ge should be used in
the synthesis of known zeolites. To plot the results in this graph, each
feature (AMD distance to a given zeolite) was normalized in a per-feature
basis. Larger SHAP values indicate higher impact in a positive classification.
Figure S13: Zeolite tree map labeled with the Pearson correlation coefficient
between AMD and SHAP values per synthesis conditions. A negative correlation
(shown in green) indicates that smaller AMD distances (i.e., high similarity)
lead to higher SHAP values (i.e., higher likelihood of a positive
classification). The correlation coefficient is the average correlation of AMD
and SHAP values for 100 XGBoost models for each synthesis condition. Figure
S14: Relationship between AMD distances computed between known and
hypothetical frameworks, and their density difference. Both known and
hypothetical zeolites were optimized with the NNPscan method by Erlenbach et
al. Figure S15: Low-dimensional projection of the hypothetical zeolite space
using their distance towards known zeolites as features. The red dots indicate
zeolites present in the IZA database. The low-dimensional plot was obtained
using UMAP (see Methods), and recovers the density and energy of the
frameworks simply by comparing them against known structures. Figure S16:
Energy-density plots for zeolites, reproducing the plot from Erlenbach et al.
The analysis of the data using these two variables shows a high concentration
of mid-energy, mid-density zeolites in the hypothetical structures database.
Figure S17: Average distance of hypothetical zeolites towards known zeolites,
visualized using the UMAP plot from Fig. S15. Brighter colors indicate lower
distances. (continues in Fig. S18). Figure S18: Average distance of
hypothetical zeolites towards known zeolites, visualized using the UMAP plot
from Fig. S15. Brighter colors indicate lower distances. (continued from Fig.
S17). Figure S19: Distributions of synthesis probabilities, as predicted by
100 XGBoost models, for all hypothetical zeolites in the “Deem dataset”.
Brighter colors indicate higher medians of the distributions.

##  Appendix B Supporting Tables

Table S1: Closest pairs of IZA zeolites according to AMDs. The pairs shown are sorted according to distances between their AMDs, and have maximum AMD distance of 0.05 Å. Zeolite 1 | Zeolite 2 | AMD distance (Å)
---|---|---
ITH | ITR | 0.016
ITG | UOV | 0.021
SBS | SBT | 0.022
ITG | IWW | 0.025
MEL | SFV | 0.032
MWF | PAU | 0.033
IMF | SFV | 0.035
IMF | TUN | 0.035
AFG | TOL | 0.037
FAR | MAR | 0.037
ERI | SWY | 0.039
OFF | SWY | 0.040
AFT | AFX | 0.040
IWW | UOV | 0.041
AFS | BPH | 0.042
SFH | SFN | 0.042
AFX | SFW | 0.044
EMT | FAU | 0.045
LIO | TOL | 0.045
PTY | PWO | 0.046
IWS | SOV | 0.047
AWO | UEI | 0.047
ITG | UWY | 0.050
Table S2: Hyperparameters explored for the logistic regression models. For the $L_{1}$ loss, only the saga solver was used. The l1_ratio parameter is used only in the case of the $L_{1}$ loss. Parameter | Choices
---|---
penalty | [l2, l1, none]
C | [0.001, 0.01, 0.1, 1, 10, 100]
solver | [lbfgs, liblinear, sag, saga]
l1_ratio | [0.25, 0.5, 0.75, 1.0]
Table S3: Hyperparameters explored for the random forest classifiers. Parameter | Choices
---|---
n_estimators | [50, 100, 200]
max_depth | [None, 10, 20]
min_samples_split | [2, 5, 10]
min_samples_leaf | [1, 2, 4]
bootstrap | [True, False]
Table S4: Hyperparameters explored for the XGBoost classifiers. Parameter | Choices
---|---
n_estimators | [50, 100, 200]
learning_rate | [0.01, 0.1, 0.2]
max_depth | [3, 4, 5, 6]
min_child_weight | [1, 2, 3]
subsample | [0.5, 0.75, 1]
colsample_bytree | [0.5, 0.75, 1]