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Emil T. S.
Kjær‡
^{a},
Andy S.
Anker‡
^{a},
Marcus N.
Weng
^{a},
Simon J. L.
Billinge
*^{bc},
Raghavendra
Selvan
*^{de} and
Kirsten M. Ø.
Jensen
*^{a}
^{a}Department of Chemistry and Nano-Science Center, University of Copenhagen, 2100 Copenhagen Ø, Denmark. E-mail: kirsten@chem.ku.dk
^{b}Department of Applied Physics and Applied Mathematics Science, Columbia University, New York, NY 10027, USA. E-mail: sb2896@columbia.edu
^{c}Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973, USA
^{d}Department of Computer Science, University of Copenhagen, 2100 Copenhagen Ø, Denmark. E-mail: raghav@di.ku.dk
^{e}Department of Neuroscience, University of Copenhagen, 2200, Copenhagen N, Denmark

Received
16th August 2022
, Accepted 28th November 2022

First published on 28th November 2022

Structure solution of nanostructured materials that have limited long-range order remains a bottleneck in materials development. We present a deep learning algorithm, DeepStruc, that can solve a simple monometallic nanoparticle structure directly from a Pair Distribution Function (PDF) obtained from total scattering data by using a conditional variational autoencoder. We first apply DeepStruc to PDFs from seven different structure types of monometallic nanoparticles, and show that structures can be solved from both simulated and experimental PDFs, including PDFs from nanoparticles that are not present in the training distribution. We also apply DeepStruc to a system of hcp, fcc and stacking faulted nanoparticles, where DeepStruc recognizes stacking faulted nanoparticles as an interpolation between hcp and fcc nanoparticles and is able to solve stacking faulted structures from PDFs. Our findings suggests that DeepStruc is a step towards a general approach for structure solution of nanomaterials.

An approach to handle the challenges due to the information barrier in PDFs is to employ supervised machine learning (ML) methods that can learn from well-known PDF-structure pairs. In this work, we use deep generative models (DGMs). DGMs are a class of ML models that can estimate the underlying data distribution from a reasonably small set of training examples.^{16} A well-known use case of DGMs is in the generation of synthetic ‘deep-fake’ images^{17,18} based on large datasets of real images. We here train our DGM to identify new structure models by training on known chemical structures. The DGM learns the relation between PDF and atomic structure, which enables it to solve monometallic nanoparticle structures, based on PDFs it has not seen before and its learned chemical knowledge. While determining a unique structure from a PDF is not always a solvable problem, as several different structures may give rise to identical PDFs, ML methods can still learn to capture the relationship between PDF and structure and thereby push the boundaries of nanostructure solution from PDF. When there is not enough information in the PDF to provide a unique structure solution, ML methods may provide a distribution of starting models which can aid in further structure analysis.

We apply our DGM, which we refer to as ‘DeepStruc’, for structural analysis of a model system of monometallic nanoparticles (MMNPs) with seven different structure types (Fig. 1a) and demonstrate the method for both simulated and experimental PDFs. DeepStruc is generative, which means that it can be used to construct structures that are not in the training set, i.e., solve a structure from a PDF. 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 structures as an interpolation between fcc and hcp and construct new structural models based on a PDF.

The measured scattering intensities are denoted I(Q), which are corrected for incoherent scattering, fluorescence, etc. and normalized such that the total scattering structure function S(Q) is obtained.

G(r) can be interpreted as a histogram of real-space interatomic distances and the information is equivalent to that of an unassigned distance matrix (uDM). All PDF simulation parameters can be found in Section G in the ESI.† The PDFs used in this project are normalised to have max(G(r)) = 1 as illustrated in Section H in the ESI.†

To further investigate the latent space behaviour of DeepStruc, a more chemically simple and intuitive dataset was made of fcc, hcp, and stacking faulted structures. Fcc and hcp can be considered layered structures that are only differentiated by the repetition of layers within the structure. Fcc consists of a repeated ABCABC layered structure where hcp is an ABABAB layered structure. A 5 layered stacking fault structure could then be described as ABCAC, as it does not satisfy either of the fcc or hcp stacking criteria. A total of 1620 stacking fault structures were generated.

The prior NN gets the PDF as input and maps it to the low-dimensional prior distribution. The low-dimensional latent vector conditioned on the PDF is then input to the decoder, which is tasked to predict the xyz-coordinates of the structural input. During the training process, the mean squared error (MSE) between the xyz-coordinates of the input and output are computed to force the decoder to predict xyx-coordinates from the latent representations. The MSE is defined as the reconstruction loss, L_{rec}. The CVAE is trained by jointly optimizing these two loss components:

L_{CVAE} = L_{rec} + βL_{recg} |

We here use MMNP structures (Fig. 1b) as input, and condition them on their simulated PDFs (Fig. 1c). The MMNP structures span seven different structure types computed using a variety of metals to emulate the variability in bond lengths in real metallic nanoparticle samples. The structure types are 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 sizes from 5 to 200 atoms. We used 3743 MMNP structures, which were randomly split into training- (60%), validation- (20%) and testing-sets (20%). Note that the validation and test sets are derived from the same underlying data distribution as the training set, and serve as intermediaries to the actual test set which is based on the experimental PDF data. A histogram of the distribution of the seven structure types are provided in Section A in the ESI.† During the training process (blue + green region Fig. 1a), DeepStruc learns to map the conditioning PDFs to their structures in the latent space. After the training process is complete, DeepStruc can be used on data that have not been part of the training set, which is referred to as ‘inference’. Further details about the DeepStruc network can be found in the Method section.

Fig. 3 Structure determination from PDFs. Simulated PDFs (grey) from the original structures of the seven different structure types (left) are used during inference for structure prediction (right). The middle column shows the fitted PDFs of the predicted structures to the simulated PDFs of the original structures. Only the scale-factor, contraction/expansion-factor, and ADP are refined, see Section B in the ESI.† |

Having established that DeepStruc works for structures highly resembling those in the training set, we now consider more challenging cases and explore the capabilities of DeepStruc on an actual test set which is far from the training distribution. As described above, the largest structures in the training set contained only 200 atoms.

We now evaluate it on a test set of simulated MMNPs with 5 to 1000 atoms, i.e., containing much larger particles. The latent space obtained from this new test set is plotted using diamond markers in Fig. 4, where the latent space from the training process is shown with semi-transparent markers. We observe that the trends in the training area are comparable for the training set and the test set of larger MMNPs. Notably, the trends of both the size and the structure types continue beyond the training area to structures containing about 400 atoms. Beyond 400 atoms, all structure types collapse onto a line, however, DeepStruc still provides a size estimate of the structure. Of course, DeepStruc could be retrained on a larger training set if reconstructions are desired on clusters larger than 200 atoms. However, this experiment shows that DeepStruc can extrapolate significantly in the latent space. It can thereby give useful information about PDFs from structures not represented in the training set and is generative in a meaningful way. This can be compared to, for example, a tree-based ML-classifier, which is limited to a predefined structural database and cannot extrapolate. The capability of DeepStruc to extrapolate arises from each structure in the latent space being predicted as a normal distribution instead of a discrete point.

Fig. 4 DeepStruc applied on PDFs of structures up to 1000 atoms. Each point is coloured after its structure type, i.e. fcc (light blue), octahedral (dark grey), decahedral (orange), bcc (green), icosahedral (dark blue), hcp (pink), and sc (red). Each point in the latent space corresponds to a structure based on its simulated PDF. Test PDFs from structures up to 1000 atoms are plotted as diamond markers on top of the training and validation data which are made semi-transparent. Note that the training set latent space is identical to that plotted in Fig. 2. DeepStruc has only been trained on structures up to 200 atoms. Three experimental PDFs (shown in Section C in the ESI†) obtained from differently sized fcc nanocrystals estimated to contain 203 (cross marker 1), 371 (cross marker 2), and 1368 (cross marker 3) atoms are illustrated as purple cross markers in the latent space. |

In practice, DeepStruc must be able to yield valid reconstructed structures from experimental data that contain noise and other aberrations. We therefore use DeepStruc to infer structures from previously published experimental PDFs from MMNPs. Fig. 5a shows the latent space with the predicted location of structures from three experimental PDFs. Here, the location in the latent space is represented as distributions rather than as discrete points, and multiple structures are sampled from each distribution and compared to the experimental PDF to select the best candidate. The mean of the experimental PDF distributions is represented as a black diamond with three ellipsoids indicating different confidence intervals with σ: 3, 5 and 7, where σ is the standard deviation of the normal distribution.

Fig. 5 Fitting experimental PDFs with structures obtained by DeepStruc. (a) The DeepStruc latent space showing predicted latent space positions for structures from three experimental PDFs. The predicted means are shown as diamond markers, which are enclosed by three rings, indicating the sampling regions for σ: 3, 5, and 7. (b) PDF fit of the reconstructed structure from the Au_{144}(p-MBA)_{60} PDF^{36} (c) PDF fit of the reconstructed structure from the 1.8 nm Pt nanoparticle PDF from Quinson et al.,^{38} (d) PDF fit of the reconstructed structure from the Au_{144}(PET)_{60} PDF^{36} using a hcp structure. (e) PDF fit of the reconstructed structure from the Au_{144}(PET)_{60} PDF^{36} using an icosahedral structure. Note that the test set structures shown here are the predicted structures from DeepStruc obtained during inference on experimental PDFs. |

The first experimental dataset that we evaluate was published by Jensen et al.,^{36} who identified a decahedral structure as the core motif of Au_{144}(p-MBA)_{60} nanoparticles. DeepStruc locates the Au_{144}(p-MBA)_{60} PDF (Fig. 5b) in a decahedral region (orange distributions in Fig. 5a) in the latent space. Given the generative capabilities of DeepStruc, in theory, we can sample an unlimited number of structures for a given PDF. As described in Section D of the ESI,† we here sampled up to 1000 structures from the three normal distributions (σ: 3, 5, and 7), and compared their fit to the experimental PDF. Fig. 5b shows the fit of the best structural prediction, which was among the structures sampled from the σ: 3 distributions. DeepStruc predicts a decahedral structure, which agrees well with the literature.^{36} Other structures sampled from the three distributions are shown in Section E of the ESI,† where we also compare the DeepStruc analysis to baseline methods. We first consider a brute-force structure-mining method inspired by Banerjee et al.,^{37} but also compare the DeepStruc results to two simpler ML-algorithms, namely a tree-based ML classifier and a regular CVAE without a graph-based input.

The second dataset that we evaluate, published by Quinson et al.,^{38} are from 1.8 nm Pt nanoparticles with the fcc structure (described further in Section C in the ESI†). This size corresponds to ca. 203 atoms, i.e. the number of atoms in the particle goes slightly beyond the fcc structures in the training set that contain only 165 atoms.^{38} The location of the predicted mean is again shown as a black diamond in Fig. 5a, enclosed by three blue ellipsoids illustrating different magnitudes of standard deviation. The mean of the predicted structure is placed near the largest sc structures. If DeepStruc only favoured symmetry it would be placed directly on the fcc structures. Interestingly, DeepStruc does not purely favour size either, as it does not position the PDF near the largest structures which are hcp structures of 200 atoms. Instead, we observe that DeepStruc takes both symmetry and size into account by placing the mean predicted structure adjacent to the largest sc structures containing 185 atoms. To identify the structure from the experimental PDF, we again sample 1000 structures from the σ: 3, 5 and 7 distributions. When fitting these sampled structures to the dataset, we obtain the best fit from an fcc structure of 146 atoms that is visualized in Fig. 5c and which agrees with the baseline models (Section E in the ESI†). DeepStruc thus identifies an fcc structure even though the size of the MMNP is outside the training set distribution.

We also attempted to input PDFs from even larger fcc nanoparticles, estimated to have diameters of 2.2 and 3.4 nm, corresponding to 371 and 1368 atoms, respectively (Section C in the ESI†).^{38} Their positions in the latent space are shown in Fig. 4 along with the 1.8 nm fcc nanoparticles using cross markers labelled 1, 2, and 3 for increasing size. We observe that they follow the trend of the simulated fcc structures discussed above: while it is possible to estimate both size and symmetry for the 2.2 nm particles through extrapolation, DeepStruc can only estimate size for the 3.4 nm particle. We note that the size can be read from a PDF directly without any modelling. However, the ability of DeepStruc to predict structures on experimental data beyond those in the training set is promising for future structure solution from PDF.

While DeepStruc only has been trained on simple MMNPs, we finally evaluate it on a PDF from Au_{144}(PET)_{60} nanoparticles, consisting of an icosahedral core of 54 atoms surrounded by a rhombicosidodecahedron shell of 60 atoms (Fig. 5d and e).^{36,39} We show the predicted mean position of the structure with a black diamond enclosed by pink ellipsoids. DeepStruc positions the PDF in the hcp 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). Similar structures are found when sampling from the σ: 3 and σ: 5 distributions. However, the PDF fit reveals that the reconstructed structure does not capture all peaks in the experimental PDF. When considering further the latent space, icosahedral structures are strongly underrepresented in our dataset (Section A in the ESI†) which results in an inconsistency when placing icosahedral structures in the latent space. DeepStruc is thus challenged when solving the icosahedral core structure of the nanoparticle. However, we observe that one of the test icosahedral structures is placed near the experimental PDF in latent space within the σ: 5 distribution. Therefore, we again try to sample 1000 structures by moving the mean of the σ: 3 distribution to the nearest cluster of icosahedral structures in the latent space, which are located right outside the σ: 7 distribution. The best fitting structure (Fig. 5e) captures all main peaks of the experimental PDF. Strategies for sampling of underrepresented structures is discussed further in Section D in the ESI.†

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. 6a. The MMNPs cluster in size, whilst we also observe that fcc and hcp structures separate in the latent space. It is evident that the stacking faulted structures are located in between the fcc and hcp structures in the latent space as hypothesized. It is chemically reasonable that they are positioned in this exact order based on their similarity to fcc and hcp. For example, the structure with ABCABA layers, shown in Fig. 6 with a purple star is structurally close fcc. We see that it is also located closer to the fcc structures in the latent space. On the other hand, the structure with ABCBCB layers (marked as a purple diamond in Fig. 6) can be considered structurally more closely related to hcp than fcc. DeepStruc places this structure adjacent to hcp structures of the same size in the latent space. DeepStruc can thus insert stacking faulted structures between fcc and hcp into the latent space in a chemically meaningful way.

Fig. 6b illustrates the fits of the reconstructed structures to the PDF data. The difference curves indicate that the predicted and true structures are very close to being identical, which is supported by the MAE of the atomic positions on 0.030 ± 0.019 Å (Section F in the ESI†). While disorder causes a broadening of the peaks, the disorder in the generated structures is minor and structures with distinct difference between the layers and in the correct sequence can be reconstructed to a satisfying degree. This is a promising result, showing that a graph-based CVAE can be used as a tool to determine the structure of stacking faulted nanoparticles from PDFs,^{41,42} which is a topic of significant current interest.^{43–47}

Our approach is only restricted by the distribution of the structural training set. When DeepStruc is trained on fcc, hcp, and stacking faulted structures, it will locate the stacking faulted structures in between the fcc and hcp structures. This suggests a strategy for training DeepStruc models on different chemical systems that also ‘interpolate’ from one to another when this can be identified. DeepStruc does not yet provide a completely general structure solution approach, but gives critical insight into how DGMs can interact with structural and diffraction information to yield candidate structures and ultimately structure solutions.

We plan to implement DeepStruc as part of PDF-in-the-cloud (https://PDFitc.org),^{48} where the training data can gradually be expanded over time. So far, the structures investigated are fairly ordered and contain some symmetry, but in the future, we plan to expand DeepStruc to chemical systems with more atoms and higher complexity such as metal oxide nanoparticles and alloys. Combining the PDF conditioning with data from complimentary techniques could prove important for structure determination of more complex systems. Such studies would both enable structure determination from a combined modelling perspective, but it would also reveal fundamental aspects of the information content of the different datasets for solving structure problems.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2dd00086e |

‡ Both authors contributed equally to this work. |

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