Original: DeepStruc: Towards structure solution from pair distribution function data using deep generative models
Revised: DeepStruc: Towards structure solution from pair distribution function data using deep generative models

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Structure solution of nanostructured materials that have limited long-range orderremains a bottleneck in materials development.
Revised: Structure solution of nanostructured materials that have limited long-range orderremains a bottleneck in materials development.

Original: We present a deep learning algorithm, DeepStruc, that can solve a simple nanoparticle structure directly from a
Revised: We present a deep learning algorithm, DeepStruc that can solve a simple nanoparticle structure directly from a

Original: Pair Distribution Function (PDF) obtained from total scattering data by using aconditional variational autoencoder.
Revised: Pair Distribution Function (PDF) obtained from total scattering data by using aconditional variational autoencoder.

Original: We first apply DeepStruc to PDFs from sevendifferent structure types of monometallic nanoparticles, and show that structurescan be solved from both simulated and experimental PDFs, including PDFs fromnanoparticles that are not present in the training distribution.
Revised: We first apply DeepStruc to PDFs from sevendifferent structure types of monometallic nanoparticles, and show that structurescan be solved from both simulated and experimental PDFs, including PDFs fromnanoparticles that are not present in the training distribution.

Original: We also apply DeepStruc to a system of hcp , fcc and stacking faulted nanoparticles, where DeepStrucrecognizes stacking faulted nanoparticles as an interpolation between hcp and fccnanoparticles and is able to solve stacking faulted structures from PDFs.
Revised: We also apply DeepStruc to a system of hcp , fcc and stacking faulted nanoparticles, where DeepStrucrecognizes stacking faulted nanoparticles as an interpolation between hcp and fccnanoparticles and is able to solve stacking faulted structures from PDFs.

Original: Ourfindings suggests that DeepStruc is a step towards a general approach for structuresolution of nanomaterials.
Revised: Ourfindings suggests that DeepStruc is a step towards a general approach for structuresolution of nanomaterials.

Original:  Introduction
Revised:  Introduction

Original: Crystallographic methods, such as single crystal and powder diffraction, have been foundational inthe development of functional materials over the past century.
Revised: Crystallographic methods, such as single crystal and powder diffraction, have been foundational inthe development of functional materials over the past century.

Original: They yield atomic-scale structuralmodels for crystalline materials and allow establishing the links between material structure andproperties that are at the heart of materials development.[1,2] However, other approaches for structuredetermination are needed for nanostructured materials that have limited long-range order and totalscattering methods such as atomic pair distribution function (PDF) analysis have become increasinglyimportant tools.[3-7]
Revised: They yield atomic-scale structuralmodels for crystalline materials and allow establishing the links between material structure andproperties that are at the heart of materials development.[1,2] However, other approaches for structuredetermination are needed for nanostructured materials that have limited long-range order, and totalscattering methods such as atomic pair distribution function (PDF) analysis have become increasinglyimportant tools.[3-7]

Original: Currently, PDF analysis is mainly done by fitting a known starting model toan experimental PDF, a process known as structure refinement.
Revised: Currently, PDF analysis is mainly done by fitting a known starting model toan experimental PDF, a process known as structure refinement.

Original: Recent developments in automatedmodelling[8-10] have made it possible to extend the searched structural space but identifying amodel or solving a structure de novo from a PDF is still an enormous challenge.
Revised: Recent developments in automatedmodelling[8-10] have made it possible to extend the searched structural space, but identifying amodel or solving a structure de novo from a PDF is still an enormous challenge.

Original: So far, only highlysymmetrical nanostructures such as the C 60 buckyball have been solved ab initio from a PDF.[11-15]
Revised: So far, only highlysymmetrical nanostructures such as the C 60 buckyball have been solved ab initio from a PDF.[11-15]

Original: Determining the structure of less symmetrical nanostructures is limited by the lost information causedby PDF peak overlap, which challenges the use of PDF for structure solution of more complicatednanomaterials.
Revised: Determining the structure of less symmetrical nanostructures is limited by the lost information causedby PDF peak overlap, which challenges the use of PDF for structure solution of more complicatednanomaterials.

Original: An approach to handle the challenges due to the information barrier in PDFs is to employ supervisedmachine learning (ML) methods that can learn from well-known PDF-structure pairs.
Revised: An approach to handle the challenges due to the information barrier in PDFs is to employ supervisedmachine learning (ML) methods that can learn from well-known PDF-structure pairs.

Original: While determining a unique structure from a PDF is not always a solvable problem, as several different structuresmay give rise to identical PDFs, ML methods can still learn to capture the relationship between PDFand structure and thereby push the boundaries of nanostructure solution from PDF.
Revised: While determining a unique structure from a PDF is not always a solvable problem, as several different structuresmay give rise to identical PDFs, ML methods can still learn to capture the relationship between PDFand structure and thereby push the boundaries of nanostructure solution from PDF.

Original: When there is notenough information in the PDF to provide a unique structure solution, ML methods may provide adistribution of starting models which can aid in further structure analysis.
Revised: When there is notenough information in the PDF to provide a unique structure solution, ML methods may provide adistribution of starting models which can aid in further structure analysis.

Original: In this work, we use deep generative models (DGMs).
Revised: In this work, we use deepgenerative models (DGMs).

Original: DGMs are a class of ML models that can estimate the underlying datadistribution from a reasonably small set of training examples.[16]
Revised: DGMs are a class of ML models that can estimate the underlying datadistribution from a reasonably small set of training examples.[16]

Original: A well-known use case of DGMsis in the generation of synthetic ‘deep-fake’ images[17,18] based on large datasets of real images.
Revised: A well-known use case of DGMsis in the generation of synthetic ‘deep-fake’ images[17,18] based on large datasets of real images.

Original: Wehere train our DGM to identify new structure models by training on known chemical structures.
Revised: Wehere train our DGM to identify new structure models by training on known chemical structures.

Original: The
Revised: The

Original: DGM learns the relation between PDF and atomic structure, which enables it to solve a structure,based on a PDF it has not seen before and its learned chemical knowledge.
Revised: DGM learns the relation between PDF and atomic structure, which enables it to solve a structure,based on a PDF it has not seen before and its learned chemical knowledge.

Original: We apply our DGM, which we refer to as ‘DeepStruc’, for structural analysis of a model system ofmonometallic nanoparticles (MMNPs) with seven different structure types (Fig. 1a) and demonstratethe method for both simulated and experimental PDFs.
Revised: We apply our DGM, which we refer to as ‘DeepStruc’, for structural analysis of a model system ofmonometallic nanoparticles (MMNPs) with seven different structure types (Fig. 1a) and demonstratethe method for both simulated and experimental PDFs.

Original: DeepStruc is generative, which means that itcan be used to construct structures that are not in the training set, i.e., solve a structure from a PDF.
Revised: DeepStruc is generative, which means that itcan be used to construct structures that are not in the training set, i.e., solve a structure from a PDF.

Original: We demonstrate this capability on a dataset of face-centered cubic ( fcc ), hexagonal closed packed( hcp ) and stacking faulted structures, where DeepStruc can recognize the stacking faulted structuresas an interpolation between fcc and hcp and construct new structural models based on a PDF.
Revised: We demonstrate this capability on a dataset of face-centered cubic ( fcc ), hexagonal closed packed( hcp ) and stacking faulted structures, where DeepStruc can recognize the stacking faulted structuresas an interpolation between fcc and hcp and construct new structural models based on a PDF.

Original:  Results2.1 Training DeepStruc to determine the structure of MMNPs from PDF data
Revised:  Results

Revised: 2.1 Training DeepStruc to determine the structure of MMNPs from PDF data

Original: DeepStruc, illustrated in Fig.
Revised: DeepStruc, illustrated in Fig.

Original: 1a and discussed below, is a graph-based conditional variational autoencoder (graph CVAE).
Revised: 1a and discussed below, is a graph-based conditional variational autoencoder (graph CVAE).

Original: Autoencoders are a class of deep learning (DL) methods where high-dimensionalinputs, such as chemical structures,[19,20] are reduced in dimensionality.
Revised: Autoencoders are a class of deep learning (DL) methods where high-dimensionalinputs, such as chemical structures,[19,20] are reduced in dimensionality.

Original: The transformation into 2or 3 dimensional vectors is achieved using an information bottleneck by an encoder neural network(NN),[19,21,22] and the resulting lower-dimensional, compressed feature space is known as the latentspace.
Revised: The transformation into 2or 3 dimensional vectors is achieved using an information bottleneck by an encoder neural network(NN),[19,21,22] and the resulting lower-dimensional, compressed feature space is known as the latent space.

Original: A decoder NN can reconstruct the input from these low-dimensional representations.
Revised: A decoder NN can reconstruct the input from these low-dimensional representations.

Original: Whenthe latent space is regularized (smoothed) using normal distributions instead of discrete points weobtain a variational autoencoder (VAE).
Revised: Whenthe latent space is regularized (smoothed) using normal distributions instead of discrete points weobtain a variational autoencoder (VAE).

Original: It has previously been demonstrated that VAEs does a betterjob interpolating in the latent space compared to deterministic AEs.[19]
Revised: It has previously been demonstrated that VAEs does a betterjob interpolating in the latent space compared to deterministic AEs.[19]

Original: The VAE can be made to bedependent (conditioned) on additional information by the prior NN resulting in a CVAE.[22]
Revised: The VAE can be made to bedependent (conditioned) on additional information by the prior NN resulting in a CVAE.[22]

Original: We here use MMNP structures (Fig. 1b) as input, and condition them on their simulated PDFs (Fig.1c).
Revised: We here use MMNP structures (Fig. 1b) as input, and condition them on their simulated PDFs (Fig.1c).

Original: The MMNP structures span seven different structure types computed using a variety of metalsto emulate the variability in bond lengths in real metallic nanoparticle samples.
Revised: The MMNP structures span seven different structure types computed using a variety of metalsto emulate the variability in bond lengths in real metallic nanoparticle samples.

Original: The structure typesare simple cubic ( sc ), body-centered cubic ( bcc ), face-centered cubic ( fcc ), hexagonal closed packed( hcp ), decahedral, icosahedral, and octahedral, and all structure types have been constructed in sizesfrom 5 to 200 atoms.
Revised: The structure typesare simple cubic ( sc ), body-centered cubic ( bcc ), face-centered cubic ( fcc ), hexagonal closed packed( hcp ), decahedral, icosahedral, and octahedral, and all structure types have been constructed in sizesfrom 5 to 200 atoms.

Original: We used 3743 MMNP structures, which were randomly split into training-(60 %), validation-
Revised: We used 3743 MMNP structures, which were randomly split into training-(60 %), validation-

Original: (20 %) and testing-sets (20 %).
Revised: (20 %) and testing-sets (20 %).

Original: Note that the validation and test sets are derivedfrom the same underlying data distribution as the training set, and serve as intermediaries to theactual test set which is based on the experimental PDF data.
Revised: Note that the validation and test sets are derivedfrom the same underlying data distribution as the training set, and serve as intermediaries to theactual test set which is based on the experimental PDF data.

Original: A histogram of the distribution ofthe seven structure types are provided in section A in the Supplementary Information.
Revised: A histogram of the distribution ofthe seven structure types are provided in section A in the Supplementary Information.

Original: During thetraining process (blue + green region Fig.
Revised: During thetraining process (blue + green region Fig.

Original: 1a)
Revised: 1a)

Original: , DeepStruc learns to map the conditioning PDFs totheir structures in the latent space.
Revised: , DeepStruc learns to map the conditioning PDFs totheir structures in the latent space.

Original: After the training process is complete, DeepStruc can be used ondata that have not been part of the training set, which is referred to as ‘inference’.
Revised: After the training process is complete, DeepStruc can be used ondata that have not been part of the training set, which is referred to as ‘inference’.

Original: Further detailsabout the DeepStruc network can be found in section B in the Supplementary Information.
Revised: Further detailsabout the DeepStruc network can be found in section B in the Supplementary Information.

Original: 2.2 Mapping of structures in a latent space
Revised: 2.2 Mapping of structures in a latent space

Original: We first evaluate DeepStruc’s ability to map the MMNP structures in a low-dimensional latent spaceby investigating structural trends and clustering.
Revised: We first evaluate DeepStruc’s ability to map the MMNP structures in a low-dimensional latent spaceby investigating structural trends and clustering.

Original: Fig.
Revised: Fig.

Original: 2 shows a visualization of the two-dimensionallatent space with selected MMNP reconstructions indicated.
Revised: 2 shows a visualization of the two-dimensionallatent space with selected MMNP reconstructions indicated.

Original: The colour of the points indicates thestructure type, and the relative point size indicates the size of the MMNP cluster.
Revised: The colour of the points indicates thestructure type, and the relative point size indicates the size of the MMNP cluster.

Original: We observe that
Revised: We observe that

Original: DeepStruc learns to map the chemical structures in the latent space by size and symmetry.
Revised: DeepStruc learns to map the chemical structures in the latent space by size and symmetry.

Original: It maps thecubic structure types ( sc , bcc , and fcc ) together, and it learns that the octahedral MMNPs are closelyrelated to the fcc structure type.
Revised: It maps thecubic structure types ( sc , bcc , and fcc ) together, and it learns that the octahedral MMNPs are closelyrelated to the fcc structure type.

Original: Interestingly, DeepStruc also allocates the decahedral structures tobe in between the fcc and hcp structures.
Revised: Interestingly, DeepStruc also allocates the decahedral structures tobe in between the fcc and hcp structures.

Original: This can be rationalized by considering that decahedralstructures are constructed from five tetrahedrally shaped fcc crystals which are separated by {111}twin boundaries that resemble stacking faults.[9,23,24]
Revised: This can be rationalized by considering that decahedralstructures are constructed from five tetrahedrally shaped fcc crystals which are separated by {111}twin boundaries that resemble stacking faults.[9,23,24]

Original: The twin boundaries will resemble stackingfaulted regions of fcc justifying that they exist in the latent space between fcc and hcp .
Revised: The twin boundaries will resemble stackingfaulted regions of fcc justifying that they exist in the latent space between fcc and hcp .

Original: 2.3 DeepStruc for structure determination from PDF
Revised: 2.3 DeepStruc for structure determination from PDF

Original: We now move on to identify structures directly from a PDF.
Revised: We now move on to identify structures directly from a PDF.

Original: The results of using DeepStruc on sevensimulated PDFs of MMNPs not used in the training process are illustrated in Fig 3.
Revised: The results of using DeepStruc on sevensimulated PDFs of MMNPs not used in the training process are illustrated in Fig 3.

Original: Here, we showthe structure that the input PDF was calculated from (left), the reconstructed structure (right), andits agreement with the input PDF after structure refinement (middle, discussed below).
Revised: Here, we showthe structure that the input PDF was calculated from (left), the reconstructed structure (right), andits agreement with the input PDF after structure refinement (middle, discussed below).

Original: In all sevencases, the structures are correctly reconstructed from the PDF input.
Revised: In all sevencases, the structures are correctly reconstructed from the PDF input.

Original: Before structure refinement, themean absolute error (MAE) of the atom positions is 0.128 ± 0.073 Å as described in section C in the
Revised: Before structure refinement, themean absolute error (MAE) of the atom positions is 0.128 ± 0.073 Å as described in section C in the

Original: Supplementary Information.
Revised: Supplementary Information.

Original: However, the MAE is artificially high due to a common aberration by
Revised: However, the MAE is artificially high due to a common aberration by

Original: DeepStruc, where it predicts the right geometric atomic arrangement, but isotropically contractedor expanded compared to the original structure.
Revised: DeepStruc, where it predicts the right geometric atomic arrangement, but isotropically contractedor expanded compared to the original structure.

Original: We do not yet understand why DeepStruc has thisaberration, but it is easily solvable by refining an expansion/contraction variable as a post processingstep to DeepStruc.
Revised: We do not yet understand why DeepStruc has thisaberration, but it is easily solvable by refining an expansion/contraction variable as a post processingstep to DeepStruc.

Original: After refining the structure to the PDF[25] by fitting a contraction/expansionfactor, a scale factor and an isotropic atomic displacement parameter (ADP), as described in section
Revised: After refining the structure to the PDF[25] by fitting a contraction/expansionfactor, a scale factor and an isotropic atomic displacement parameter (ADP), as described in section

Original: C in the Supplementary Information, the MAE of the atom positions is reduced to 0.093 ± 0.058Å.
Revised: C in the Supplementary Information, the MAE of the atom positions is reduced to 0.093 ± 0.058Å.

Original: The inference is thus robust against moderate changes in lattice parameter between a provided
Revised: The inference is thus robust against moderate changes in lattice parameter between a provided

Original: PDF and the structures that DeepStruc were trained on.
Revised: PDF and the structures that DeepStruc were trained on.

Original: The reconstructed structures exhibit some artificial positional atomic disorder that broadens the PDF peaks.
Revised: The reconstructed structures exhibit someartificial positional atomic disorder that broadens the PDF peaks.

Original: The fitted ADP values (section C inthe Supplementary Information) are thus lower than the ADP values of the conditioning PDFs.
Revised: The fitted ADP values (section C inthe Supplementary Information) are thus lower than the ADP values of the conditioning PDFs.

Original: Having established that DeepStruc works for structures highly resembling those in the training set, wenow consider more challenging cases and explore the capabilities of DeepStruc on an actual test setwhich is far from the training distribution.
Revised: Having established that DeepStruc works for structures highly resembling those in the training set, wenow consider more challenging cases and explore the capabilities of DeepStruc on an actual test setwhich is far from the training distribution.

Original: As described above, the largest structures in the trainingset contained only 200 atoms.
Revised: As described above, the largest structures in the trainingset contained only 200 atoms.

Original: We now evaluate it on a test set of simulated MMNPs with 5 to 1000 atoms, i.e., containing muchlarger particles.
Revised: We now evaluate it on a test set of simulated MMNPs with 5 to 1000 atoms, i.e., containing muchlarger particles.

Original: The latent space obtained from this new test set is plotted using diamond markers in
Revised: The latent space obtained from this new test set is plotted using diamond markers in

Original: Fig.
Revised: Fig.

Original: 4, where the latent space from the training process is shown with semi-transparent markers.
Revised: 4, where the latent space from the training process is shown with semi-transparent markers.

Original: Weobserve that the trends in the training area are comparable for the training set and the test set of larger
Revised: Weobserve that the trends in the training area are comparable for the training set and the test set of larger

Original: MMNPs.
Revised: MMNPs.

Original: Notably, the trends of both the size and the structure types continue beyond the training areato structures containing about 400 atoms.
Revised: Notably, the trends of both the size and the structure types continue beyond the training areato structures containing about 400 atoms.

Original: Beyond 400 atoms, all structure types collapse onto a line,however, DeepStruc still estimates the size of the structure.
Revised: Beyond 400 atoms, all structure types collapse onto a line,however, DeepStruc still estimates the size of the structure.

Original: Of course, DeepStruc could be retrainedon a larger training set if reconstructions are desired on clusters larger than 200 atoms.
Revised: Of course, DeepStruc could be retrainedon a larger training set if reconstructions are desired on clusters larger than 200 atoms.

Original: However, thisexperiment shows that DeepStruc can extrapolate significantly in the latent space.
Revised: However, thisexperiment shows that DeepStruc can extrapolate significantly in the latent space.

Original: It can thereby giveuseful information about PDFs from structures not represented in the training set and is generative ina meaningful way.
Revised: It can thereby giveuseful information about PDFs from structures not represented in the training set and is generative ina meaningful way.

Original: This can be compared to, for example, a tree-based ML-classifier, which is limitedto a predefined structural database and cannot extrapolate.
Revised: This can be compared to, for example, a tree-based ML-classifier, which is limitedto a predefined structural database and cannot extrapolate.

Original: The capability of DeepStruc to extrapolatearises from each structure in the latent space being predicted as a normal distribution instead of adiscrete point.
Revised: The capability of DeepStruc to extrapolatearises from each structure in the latent space being predicted as a normal distribution instead of adiscrete point.

Original: In practice, DeepStruc must be able to yield valid reconstructed structures from experimental datathat contain noise and other aberrations.
Revised: In practice, DeepStruc must be able to yield valid reconstructed structures from experimental datathat contain noise and other aberrations.

Original: We therefore use DeepStruc to infer structures frompreviously published experimental PDFs from MMNPs.
Revised: We therefore use DeepStruc to infer structures from previously published experimental PDFs from MMNPs.

Original: Fig.
Revised: Fig.

Original: 5a shows the latent space with thepredicted location of structures from three experimental PDFs.
Revised: 5a shows the latent space with thepredicted location of structures from three experimental PDFs.

Original: Here, the location in the latent spaceis represented as distributions rather than as discrete points, and multiple structures are sampled fromeach distribution and compared to the experimental PDF to select the best candidate.
Revised: Here, the location in the latent spaceis represented as distributions rather than as discrete points, and multiple structures are sampled fromeach distribution and compared to the experimental PDF to select the best candidate.

Original: The mean ofthe experimental PDF distributions is represented as a black diamond with three ellipsoids indicatingdifferent confidence intervals with σ : 3, 5 and 7, where σ is the standard deviation of the normaldistribution.
Revised: The mean ofthe experimental PDF distributions is represented as a black diamond with three ellipsoids indicatingdifferent confidence intervals with σ : 3, 5 and 7, where σ is the standard deviation of the normaldistribution.

Original: The first experimental dataset that we evaluate was published by Jensen et al.
Revised: The first experimental dataset that we evaluate was published by Jensen et al.

Original: ,[26] who identifieda decahedral structure as the core motif of Au 144 ( p -MBA) 60 nanoparticles.
Revised: ,[26] who identifieda decahedral structure as the core motif of Au 144 ( p -MBA) 60 nanoparticles.

Original: DeepStruc locates the
Revised: DeepStruc locates the

Original: Au 144 ( p -MBA) 60 PDF (Fig. 5b) in a decahedral region (orange distributions in Fig.
Revised: Au 144 ( p -MBA) 60 PDF (Fig. 5b) in a decahedral region (orange distributions in Fig.

Original: 5a) in the latentspace.
Revised: 5a) in the latentspace.

Original: Given the generative capabilities of DeepStruc, in theory, we can sample an unlimited numberof structures for a given PDF.
Revised: Given the generative capabilities of DeepStruc, in theory, we can sample an unlimited numberof structures for a given PDF.

Original: As described in section D of the Supplementary Information, we heresampled up to 1000 structures from the three normal distributions ( σ : 3, 5, and 7), and compared theirfit to the experimental PDF.
Revised: As described in section D of the Supplementary Information, we heresampled up to 1000 structures from the three normal distributions ( σ : 3, 5, and 7), and compared theirfit to the experimental PDF.

Original: Fig.
Revised: Fig.

Original: 5b shows the fit of the best structural prediction, which was amongthe structures sampled from the σ : 3 distributions.
Revised: 5b shows the fit of the best structural prediction, which was amongthe structures sampled from the σ : 3 distributions.

Original: DeepStruc predicts a decahedral structure, whichagrees well with the literature.[26] Other structures sampled from the three distributions are shownin Section E of the Supplementary Information, where we also compare the DeepStruc analysis tobaseline methods.
Revised: DeepStruc predicts a decahedral structure, whichagrees well with the literature.[26] Other structures sampled from the three distributions are shownin Section E of the Supplementary Information, where we also compare the DeepStruc analysis tobaseline methods.

Original: We first consider a brute-force structure-mining method inspired by Banerjee etal.
Revised: We first consider a brute-force structure-mining method inspired by Banerjee etal.

Original: ,[27], but also compare the DeepStruc results to two simpler ML-algorithms, namely a tree-based
Revised: ,[27], but also compare the DeepStruc results to two simpler ML-algorithms, namely a tree-based

Original: ML classifier and a regular CVAE without a graph-based input.
Revised: ML classifier and a regular CVAE without a graph-based input.

Original: The second dataset that we evaluate, published by Quinson et al.
Revised: The second dataset that we evaluate, published by Quinson et al.

Original: ,[28] are from 1.8 nm Pt nanoparticleswith the fcc structure (described further in Section F in the Supplementary Information).
Revised: ,[28] are from 1.8 nm Pt nanoparticleswith the fcc structure (described further in Section F in the Supplementary Information).

Original: This sizecorresponds to ca.
Revised: This sizecorresponds to ca.

Original: 203 atoms, i.e. the number of atoms in the particle goes slightly beyond the fccstructures in the training set that contain only 165 atoms.[28]
Revised: 203 atoms, i.e. the number of atoms in the particle goes slightly beyond the fccstructures in the training set that contain only 165 atoms.[28]

Original: The location of the predicted mean isagain shown as a black diamond in Fig.
Revised: The location of the predicted mean isagain shown as a black diamond in Fig.

Original: 5a, enclosed by three blue ellipsoids illustrating differentmagnitudes of standard deviation.
Revised: 5a, enclosed by three blue ellipsoids illustrating differentmagnitudes of standard deviation.

Original: The mean of the predicted structure is placed near the largest scstructures.
Revised: The mean of the predicted structure is placed near the largest sc structures.

Original: If DeepStruc only favoured symmetry it would be placed directly on the fcc structures.
Revised: If DeepStruc only favoured symmetry it would be placed directly on the fcc structures.

Original: Interestingly, DeepStruc does not purely favour size either, as it does not position the PDF near thelargest structures which are hcp structures of 200 atoms.
Revised: Interestingly, DeepStruc does not purely favour size either, as it does not position the PDF near thelargest structures which are hcp structures of 200 atoms.

Original: Instead, we observe that DeepStruc takesboth symmetry and size into account by placing the mean predicted structure adjacent to the largestsc structures containing 185 atoms.
Revised: Instead, we observe that DeepStruc takesboth symmetry and size into account by placing the mean predicted structure adjacent to the largestsc structures containing 185 atoms.

Original: To identify the structure from the experimental PDF, we againsample 1000 structures from the σ : 3, 5 and 7 distributions.
Revised: To identify the structure from the experimental PDF, we againsample 1000 structures from the σ : 3, 5 and 7 distributions.

Original: When fitting these sampled structures tothe dataset, we obtain the best fit from an fcc structure of 146 atoms that is visualized in Fig.
Revised: When fitting these sampled structures tothe dataset, we obtain the best fit from an fcc structure of 146 atoms that is visualized in Fig.

Original: 5c and which agrees with the baseline models (section E in the Supplementary Information).
Revised: 5c andwhich agrees with the baseline models (section E in the Supplementary Information).

Original: DeepStruc thusidentifies an fcc structure even though the size of the MMNP is outside the training set distribution.
Revised: DeepStruc thusidentifies an fcc structure even though the size of the MMNP is outside the training set distribution.

Original: We also attempted to input PDFs from even larger fcc nanoparticles, estimated to have diameters of2.2 and 3.4 nm, corresponding to 371 and 1368 atoms, respectively (section F in the Supplementary
Revised: We also attempted to input PDFs from even larger fcc nanoparticles, estimated to have diameters of2.2 and 3.4 nm, corresponding to 371 and 1368 atoms, respectively (section F in the Supplementary

Original: Information).[28]
Revised: Information).[28]

Original: Their positions in the latent space are shown in Fig.
Revised: Their positions in the latent space are shown in Fig.

Original: 4 along with the 1.8 nm fccnanoparticles using cross markers labelled 1, 2, and 3 for increasing size.
Revised: 4 along with the 1.8 nm fccnanoparticles using cross markers labelled 1, 2, and 3 for increasing size.

Original: We observe that they followthe trend of the simulated fcc structures discussed above: while it is possible to estimate both size andsymmetry for the 2.2 nm particles through extrapolation, DeepStruc can only estimate size for the 3.4nm particle.
Revised: We observe that they followthe trend of the simulated fcc structures discussed above: while it is possible to estimate both size andsymmetry for the 2.2 nm particles through extrapolation, DeepStruc can only estimate size for the 3.4nm particle.

Original: We note that the size can be read from a PDF directly without any modelling.
Revised: We note that the size can be read from a PDF directly without any modelling.

Original: However,the ability of DeepStruc to predict structures on experimental data beyond those in the training set ispromising for future structure solution from PDF.
Revised: However,the ability of DeepStruc to predict structures on experimental data beyond those in the training set ispromising for future structure solution from PDF.

Original: While DeepStruc only has been trained on simple MMNPs, we finally evaluate it on a PDF from
Revised: While DeepStruc only has been trained on simple MMNPs, we finally evaluate it on a PDF from

Original: Au 144 (PET) 60 nanoparticles, consisting of an icosahedral core of 54 atoms surrounded by a rhombicosidodecahedron shell of 60 atoms (Fig.
Revised: Au 144 (PET) 60 nanoparticles, consisting of an icosahedral core of 54 atoms surrounded by a rhombicosidodecahedron shell of 60 atoms (Fig.

Original: 5d and e).[26,29]
Revised: 5d and e).[26,29]

Original: We show the predicted mean position ofthe structure with a black diamond enclosed by pink ellipsoids.
Revised: We show the predicted mean position ofthe structure with a black diamond enclosed by pink ellipsoids.

Original: DeepStruc positions the PDF in thehcp region of the latent space, and when sampling 1000 structures from the distribution with σ : 7,the best fitting structures is an hcp structure with 40 atoms for the Au 144 (PET) 60 nanoparticle (Fig.5d).
Revised: DeepStruc positions the PDF in thehcp region of the latent space, and when sampling 1000 structures from the distribution with σ : 7,the best fitting structures is an hcp structure with 40 atoms for the Au 144 (PET) 60 nanoparticle (Fig.5d).

Original: Similar structures are found when sampling from the σ : 3 and σ : 5 distributions.
Revised: Similar structures are found when sampling from the σ : 3 and σ : 5 distributions.

Original: However, the
Revised: However, the

Original: PDF fit reveals that the reconstructed structure does not capture all peaks in the experimental PDF.
Revised: PDF fit reveals that the reconstructed structure does not capture all peaks in the experimental PDF.

Original: When considering further the latent space, icosahedral structures are strongly underrepresented in ourdataset (section A in the Supplementary Information) which results in an inconsistency when placingicosahedral structures in the latent space.
Revised: When considering further the latent space, icosahedral structures are strongly underrepresented in ourdataset (section A in the Supplementary Information) which results in an inconsistency when placingicosahedral structures in the latent space.

Original: DeepStruc is thus challenged when solving the icosahedralcore structure of the nanoparticle.
Revised: DeepStruc is thus challenged when solving the icosahedralcore structure of the nanoparticle.

Original: However, we observe that one of the test icosahedral structures isplaced near the experimental PDF in latent space within the σ : 5 distribution.
Revised: However, we observe that one of the test icosahedral structures isplaced near the experimental PDF in latent space within the σ : 5 distribution.

Original: Therefore, we againtry to sample 1000 structures by moving the mean of the σ : 3 distribution to the nearest clusterof icosahedral structures in the latent space, which are located right outside the σ : 7 distribution.
Revised: Therefore, we againtry to sample 1000 structures by moving the mean of the σ : 3 distribution to the nearest clusterof icosahedral structures in the latent space, which are located right outside the σ : 7 distribution.

Original: The best fitting structure (Fig. 5e) captures all main peaks of the experimental PDF.
Revised: The best fitting structure (Fig. 5e) captures all main peaks of the experimental PDF.

Original: Strategiesfor sampling of underrepresented structures is discussed further in section D in the Supplementary
Revised: Strategiesfor sampling of underrepresented structures is discussed further in section D in the Supplementary

Original: Information.
Revised: Information.

Original: 2.4 Structure determination from PDF: fcc , hcp , and stacking faulted nanoparticles
Revised: 2.4 Structure determination from PDF: fcc , hcp , and stacking faulted nanoparticles

Original: To obtain a deeper understanding of the latent space’s behaviour, we investigate a dataset onlycontaining fcc , hcp , and stacking faulted structures.
Revised: To obtain a deeper understanding of the latent space’s behaviour, we investigate a dataset onlycontaining fcc , hcp , and stacking faulted structures.

Original: Fcc and hcp structures are distinguished by thestacking sequence of closed packed layers in their structures: while fcc structures can be describedby ABCABC stacking, hcp structures have ABABAB stacking.
Revised: Fcc and hcp structures are distinguished by thestacking sequence of closed packed layers in their structures: while fcc structures can be describedby ABCABC stacking, hcp structures have ABABAB stacking.

Original: Structures with other sequences arestacking faulted structures.
Revised: Structures with other sequences arestacking faulted structures.

Original: We hypothesize that stacking faulted structures can be considered an‘interpolation’ in the discrete space between the fcc and hcp structure type.[30]
Revised: We hypothesize that stacking faulted structures can be considered an‘interpolation’ in the discrete space between the fcc and hcp structure type.[30]

Original: Examples of reconstructed fcc (blue), hcp (pink), and different stacking faulted structures (purple)and their position in the new latent space are illustrated in Fig.
Revised: Examples of reconstructed fcc (blue), hcp (pink), and different stacking faulted structures (purple)and their position in the new latent space are illustrated in Fig.

Original: S8a.
Revised: S8a.

Original: The MMNPs cluster in size,whilst we also observe that fcc and hcp structures separate in the latent space.
Revised: The MMNPs cluster in size,whilst we also observe that fcc and hcp structures separate in the latent space.

Original: It is evident that thestacking faulted structures are located in between the fcc and hcp structures in the latent space ashypothesized.
Revised: It is evident that thestacking faulted structures are located in between the fcc and hcp structures in the latent space ashypothesized.

Original: It is chemically reasonable that they are positioned in this exact order based on their similarity to fcc and hcp .
Revised: It is chemically reasonable that they are positioned in this exact order based on theirsimilarity to fcc and hcp .

Original: For example, the structure with ABCABA layers, shown in Fig.
Revised: For example, the structure with ABCABA layers, shown in Fig.

Original: S8 with apurple star is structurally close fcc .
Revised: S8 with apurple star is structurally close fcc .

Original: We see that it is also located closer to the fcc structures in thelatent space.
Revised: We see that it is also located closer to the fcc structures in thelatent space.

Original: On the other hand, the structure with ABCBCB layers (marked as a purple diamond in
Revised: On the other hand, the structure with ABCBCB layers (marked as a purple diamond in

Original: Fig. S8) can be considered structurally more closely related to hcp than fcc .
Revised: Fig. S8) can be considered structurally more closely related to hcp than fcc .

Original: DeepStruc places thisstructure adjacent to hcp structures of the same size in the latent space.
Revised: DeepStruc places thisstructure adjacent to hcp structures of the same size in the latent space.

Original: DeepStruc can thus insertstacking faulted structures between fcc and hcp into the latent space in a chemically meaningful way.
Revised: DeepStruc can thus insertstacking faulted structures between fcc and hcp into the latent space in a chemically meaningful way.

Original: Fig.
Revised: Fig.

Original: S8b illustrates the fits of the reconstructed structures to the PDF data.
Revised: S8b illustrates the fits of the reconstructed structures to the PDF data.

Original: The difference curvesindicate that the predicted and true structures are very close to being identical, which is supported bythe MAE of the atomic positions on 0.030 ± 0.019 Å (section E in the Supplementary Information).
Revised: The difference curvesindicate that the predicted and true structures are very close to being identical, which is supported bythe MAE of the atomic positions on 0.030 ± 0.019 Å (section E in the Supplementary Information).

Original: While disorder causes a broadening of the peaks, the disorder in the generated structures is minor andstructures with distinct difference between the layers and in the correct sequence can be reconstructedto a satisfying degree.
Revised: While disorder causes a broadening of the peaks, the disorder in the generated structures is minor andstructures with distinct difference between the layers and in the correct sequence can be reconstructedto a satisfying degree.

Original: This is a promising result, showing that a graph-based CVAE can be used as atool to determine the structure of stacking faulted nanoparticles from PDFs,[31,32] which is a topicof significant current interest.[33-37]
Revised: This is a promising result, showing that a graph-based CVAE can be used as atool to determine the structure of stacking faulted nanoparticles from PDFs,[31,32] which is a topicof significant current interest.[33-37]

Original:  Discussion
Revised:  Discussion

Original: We have shown the potential of using a DGM for structure determination from simulated andexperimental PDFs.
Revised: We have shown the potential of using a DGM for structure determination from simulated andexperimental PDFs.

Original: Our graph-based CVAE algorithm, DeepStruc, provides valuable informationthrough its latent space, as the MMNP structures cluster based on symmetry and size in agreementwith their structural chemistry.
Revised: Our graph-based CVAE algorithm, DeepStruc, provides valuable informationthrough its latent space, as the MMNP structures cluster based on symmetry and size in agreementwith their structural chemistry.

Original: Using experimental data, the Au 144 ( p -MBA) 60 nanoparticle wasdetermined to be decahedral, Pt nanoparticles were determined to be fcc and the Au 144 (PET) 60 wasdetermined to have an icosahedral core structure, all in agreement with previous literature.
Revised: Using experimental data, the Au 144 ( p -MBA) 60 nanoparticle wasdetermined to be decahedral, Pt nanoparticles were determined to be fcc and the Au 144 (PET) 60 wasdetermined to have an icosahedral core structure, all in agreement with previous literature.

Original: Whilethese systems are relatively simple MMNPs, we recognise that there are more complex materialswhere the measured PDF would not contain sufficient information to solve the structure.
Revised: While these systems are relatively simple MMNPs, we recognise that there are more complex materialswhere the measured PDF would not contain sufficient information to solve the structure.

Original: DeepStrucwould then be limited to provide a distribution of starting models which can aid in the further structureanalysis.
Revised: DeepStrucwould then be limited to provide a distribution of starting models which can aid in the further structureanalysis.

Original: Our approach is only restricted by the distribution of the structural training set.
Revised: Our approach is only restricted by the distribution of the structural training set.

Original: When DeepStrucis trained on fcc , hcp , and stacking faulted structures, it will locate the stacking faulted structuresin between the fcc and hcp structures.
Revised: When DeepStrucis trained on fcc , hcp , and stacking faulted structures, it will locate the stacking faulted structuresin between the fcc and hcp structures.

Original: This suggests a strategy for training DeepStruc models ondifferent chemical systems that also ‘interpolate’ from one to another when this can be identified.
Revised: This suggests a strategy for training DeepStruc models ondifferent chemical systems that also ‘interpolate’ from one to another when this can be identified.

Original: DeepStruc does not yet provide a completely general structure solution approach, but gives criticalinsight into how DGMs can interact with structural and diffraction information to yield candidatestructures and ultimately structure solutions.
Revised: DeepStruc does not yet provide a completely general structure solution approach, but gives criticalinsight into how DGMs can interact with structural and diffraction information to yield candidatestructures and ultimately structure solutions.

Original: We suggest to implement DeepStruc as part of PDF-in-the-cloud (PDFitc.org),[38] where the trainingdata can gradually be expanded over time.
Revised: We suggest to implement DeepStruc as part of PDF-in-the-cloud (PDFitc.org),[38] where the trainingdata can gradually be expanded over time.

Original: So far, the structures investigated are fairly ordered andcontain some symmetry, but in the future, we plan to expand DeepStruc to chemical systems withmore atoms and higher complexity such as metal oxide nanoparticles and alloys.
Revised: So far, the structures investigated are fairly ordered andcontain some symmetry, but in the future, we plan to expand DeepStruc to chemical systems withmore atoms and higher complexity such as metal oxide nanoparticles and alloys.

Original: Combining the
Revised: Combining the

Original: PDF conditioning with data from complimentary techniques could prove important for structuredetermination of more complex systems.
Revised: PDF conditioning with data from complimentary techniques could prove important for structuredetermination of more complex systems.

Original: Such studies would both enable structure determination froma combined modelling perspective, but it would also reveal fundamental aspects of the informationcontent of the different datasets for solving structure problems.  
Revised: Such studies would both enable structure determination froma combined modelling perspective, but it would also reveal fundamental aspects of the informationcontent of the different datasets for solving structure problems.  Data availability

Revised: Code for DeepStruc and the baseline models are available at:https://github.com/EmilSkaaning/DeepStruchttps://github.com/AndyNano/Brute-force-PDF-modellinghttps://github.com/AndyNano/MetalFinderhttps://github.com/AndyNano/CVAE

Revised:  Acknowledgements

Revised: Acknowledgements This work is part of a project that has received funding from the European

Revised: Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement No. 804066).

Revised: We are grateful to the Villum Foundation for financialsupport through a Villum Young Investigator grant (VKR00015416).

Revised: Funding from the Danish Ministry of Higher Education and Science through the SMART Lighthouse is gratefully acknowledged.

Revised: We acknowledge support from the Danish National Research Foundation Center for High Entropy

Revised: Alloy Catalysis (DNRF 149).

Revised: Work in the Billinge group was supported by the U.S. National Science

Revised: Foundation through grant DMREF-1922234.

Revised:  Author contributions

Revised: ETSK and ASA contributed to all aspects of the paper.

Revised: MNW wrote the code associated to thetree-based classifier.

Revised: SJLB, RS and KMØJ supervised the project.

Revised: All authors contributed to thewriting of the manuscript.

Revised:  Competing interests

Revised: Competing interests The authors declare no competing interests.