Neural network based classification of crystal symmetries from x-ray diffraction patterns

Machine learning algorithms based on artificial neural networks have proven very useful for a variety of classification problems. Here we apply them to a well-known problem in crystallography, namely the classification of x-ray diffraction (XRD) patterns of inorganic powder specimens by the respective crystal system and space group. Over ${10}^5$ theoretically computed powder XRD patterns were obtained from inorganic crystal structure databases and used to train a deep dense neural network. For space group classification, we obtain an accuracy of around 54% on experimental data. Finally, we introduce a scheme where the network has the option to refuse the classification of XRD patterns that would be classified with a large uncertainty. This enhances the accuracy on experimental data to 82% at the expense of having half of the experimental data unclassified. With further improvements of neural network architecture and experimental data availability, machine learning constitutes a promising complement to classical structure determination methodology.

Figure 1

Typical samples of powder XRD patterns. (a) XRD pattern for Borax ${\text{Na}}_2\left[{\text{B}}_4{\text{O}}_5{\text{(OH)}}_4\right]·8{\text{H}}_2\text{O}$ (obtained from the RRUFF database); (b) theoretically computed XRD pattern for ${\text{C}}_2\left({\text{Si}}_6{\text{Cl}}_{14}\right)$ including noise and background. Both compounds belong to the space group 15.

Figure 2

Network architectures used for classifying space groups from powder XRD patterns. (a) Convolutional neural network as used in Ref. [23]. (b) Dense network. Blue layers represents convolutional layers with $f$ filters of size $k$ and stride $s$, green layers correspond to dense layers with the number of nodes given by the value within and yellow layers are dropout layers with the indicated dropout rate. The gray layers are the input layers with the number denoting the corresponding input size. The XRD patterns are zero padded to match the input size. Evaluated on the experimental samples from the RRUFF database, we achieved an accuracy of $42\%$ for the convolutional network and an accuracy of $54\%$ for the dense network.

Figure 3

Network classification accuracies. (a) and (b) The evolution of the network accuracies evaluated over the test set and the RRUFF database, which consists of experimentally obtained XRD patterns, for (a) the classification of crystal systems and (b) the classification of space groups. The accuracies are averaged over 10 networks initialized with a different random seeds. While the convolutional network performs better on the training data in both cases, the dense network generalizes better to the experimental RRUFF database. (c) Histogram of the distances between network's prediction and correct space group classification. This distance is defined on the maximal subgroup graph.

Figure 4

Classification accuracy of individual space groups. The colors denote the corresponding crystal system and the area of the circle indicates its relative abundance in the training set. Space groups corresponding to larger indices tend to contain more symmetry elements. For instance, space group 1 contains no point group symmetries whereas space group 230 contains the full set of cubic symmetries. Depicted here are the accuracies for the theoretically obtained test set, because the RRUFF database is too small to provide meaningful statistics for individual space groups. (There are around 800 samples for 230 space groups in the RRUFF database.)

Figure 5

(a) Ratio of samples whose corresponding space group is among the $n$ highest outputs of the network. (b) and (c) Histograms showing the number of samples in the RRUFF database which are classified correctly or wrongly with a given certainty [defined in Eq. (13)]. One can see that when the network is more certain, it is also more often correct. (b) Convolutional network. (c) Dense network. (d) and (e) Depicts the accuracy (fraction of correctly classified samples on the RRUFF database) of the network if we restrict the network's certainty to be above a certain cutoff threshold. Given a XRD sample, if the network gives a certainty below the threshold, then the sample is ignored or unclassified. The orange line gives the ratio of unclassified samples as the cutoff is varied while the blue lines shows the accuracy of the network given a certain ratio of unclassified samples. The dotted lines indicate the accuracy when $50\%$ of the samples are left unclassified. At this cutoff level, the accuracies are $65\%$ and $82\%$ for the (d) convolutional network and the (e) dense network, respectively.

Figure 6

Confusion matrices are evaluated only on the theoretical test data (7000 samples), because the experimental RRUFF database is too small to provide meaningful statistics for individual space groups. (There are around 800 samples for 230 space groups in the RRUFF database.) The white strips in the figure corresponds to space groups which are not present among the 7000 test samples. (a) Convolutional network. (b) Dense network.