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Compositional disorder induces myriad captivating phenomena in perovskites. Target-driven discovery of perovskite solid solutions has been a great challenge due to the analytical complexity introduced by disorder. Here, we demonstrate that an unsupervised deep learning strategy can find fingerprints of disordered materials that embed perovskite formability and underlying crystal structure information by learning only from the chemical composition, manifested in \((\rmA_1-\rmx\rmA^\prime _\rmx)\rmBO_3\) and \(\rmA(\rmB_1-\rmx\rmB^\prime _\rmx)\rmO_3\) formulae. This phenomenon can be capitalized to predict the crystal symmetry of experimental compositions, outperforming several supervised machine learning (ML) algorithms. The educated nature of material fingerprints has led to the conception of analogical materials discovery that facilitates inverse exploration of promising perovskites based on similarity investigation with known materials. The search space of unstudied perovskites is screened from 600,000 feasible compounds using experimental data powered ML models and automated web mining tools at a 94% success rate. This concept further provides insights on possible phase transitions and computational modelling of complex compositions. The proposed quantitative analysis of materials analogies is expected to bridge the gap between the existing materials literature and the undiscovered terrain.
An appropriate feature space is developed starting from the chemical formula to represent each material in the experimental database and candidate pool. While characterization equipment such as X-ray photoemission spectroscopy is required to investigate the oxidation state of constituent elements28, here we implement a heuristic algorithm to unroll fractional oxidation states, closely reflecting the actual cation valences. We further add ten best perovskite descriptors procured from the sure independence screening and sparsifying operator (SISSO) algorithm11,29 to the feature space, consolidating hand-crafted and data-driven descriptors. A sequential screening strategy that involves ML classification, chemistry informed filtering and automated web scraping is employed to downselect promising unknown perovskites with 93.9% fidelity. We construct a variational autoencoder (VAE) model30 and train it on a sufficiently large, unlabelled experimental database, sanguinely retrieving a low-dimensional representation of compositions, designated as the material fingerprint (Fig. 1d). Notably, VAEs have been purposed as molecule generative models31 and feature extractors where the latent vector is sent through another ML model to predict element combinations that will likely form a specific topology, requiring crystallographic data as the input32. VAE-obtained latent vector has also been used to predict optical properties of materials33. Here, we demonstrate the exclusive potential of material fingerprints learned only from the chemical composition to carry crystal structure and perovskite formability information as numerical vectors, stressing their applicability in analogical materials discovery. Material attributes such as bandgap and phase transition details can be recovered by proximity analysis of fingerprint topology assembled by the VAE. Through a series of statistical testing on crystal symmetry prediction, we illustrate that structurally and physiochemically similar materials likely possess similar fingerprint representations. This phenomenon is employed to discover lead-free analogues of several lead-based ceramics. We further exemplify the versatility of material fingerprints in initializing the DFT simulation of disordered solid solutions by modelling six promising lead-free perovskites (see Fig. 1).
The roadmap to analogical materials discovery begins by establishing a solid and adequate collection of candidate compounds for the parent experimental material or class of materials. We use pymatgen34, an open-source python library for analyzing the materials in our database. Pymatgen is equipped to guess the most probable oxidation states of elements in a composition based on ICSD statistics. In the case of mixed valence constituent element, this may lead to a fractional oxidation state that must be unfolded to estimate the effective ionic radius of that element and the associated crystallographic site in the average unit cell. Considering the smallest possible supercell that relates to the integer formula, our algorithm first tries to solve for the valence states of individual ions by exploiting the common oxidation states of the mixed-valence element in question. If no solution is found, it looks through other less common oxidations states. Whenever the algorithm finds more than one configuration of valence states can result in the same fractional oxidation state, the one with the smallest oxidation state numbers and the lowest variation in ionic radii is selected (see Supplementary Note 2). We expect the perovskite structure is more likely to be retained under these conditions because, if there is large difference in size between the constituent cations, the octahedral units share faces, distorting the perovskite lattice more6. However, it should be noted that mixed valence states are highly dependent on the element, composition, vacancies etc., requiring experimental characterization for precise investigation. Our algorithm provides a systematic estimation of the mean ionic radius of such elements, permitting us to calculate t, μ and develop the feature space. Subsequently, t and μ of \((\rmA_1-\rmx\rmA^\prime _\rmx)\rmBO_3\) type compositions are computed as1;
We examined the performance of four ML algorithms in the present classification task to select the best model to proceed. These include decision tree (DT), random forest (RF), support vector machine (SVM) and gradient boosting classifier (GBC). In order to assess the performance metrics, we iteratively trained and evaluated the individual models on sub-databases generated by integrating 227 non-perovskites with 250 randomly sampled perovskites from 1758 total perovskites. Such a strategy is required to maintain the class-balance of individual training sets, minimizing the potential overfitting to a particular class. It also ensures that all available perovskites are exploited without being dissipated. All four algorithms are trained and validated for 100 independent iterations by splitting each sub-database into 80% training and 20% test sets19. Table 1 summarizes tenfold cross validation (CV) results. GBC is selected to proceed as it provided the best performance with 93.91% (0.96%) and 93.33% (2.26%) tenfold CV and test mean classification accuracies, respectively. More details about these metrics including hyperparameter tuning are available in Methods, and Supplementary Fig. 1 and Supplementary Note 4. Figure 2b,c depict confusion matrix and receiver operating characteristic (ROC) curves, respectively, as obtained by tenfold CV. Moreover, five SISSO descriptors make it to the top ten features ranked by their relative importance (Fig. 2a). However, we note that the GBC model achieves 93.71% and 93.16% tenfold CV and test mean accuracies, respectively, even in the absence of SISSO descriptors in the feature space, endorsing the fact that perovskite formability is heavily dependent on geometrical features such as cationic radii.
Each panel depicts the classification probability variation as the content x of different \(\rmB^\prime\) cations increases, for some commonly observed A and B element combinations. Empty spaces imply no charge balanced probable compositions exist. Caesium cation is too large to occupy A-site and the formation of a stable perovskite is generally improbable. ICSD reports one quaternary perovksite oxide, K(Ta0.77Nb0.23)O3, with potassium fully occupying A-site, as opposed to four stable ternary perovskites, KNbO3, KTaO3, KUO3 and KIO3. Despite having no training data with A-site K and B-site U occupied perovskites, the model predicts that U can be alloyed with Nb, Ta or even Ti and Zr to forge a perovskite. This claim is somewhat supported by above ternary oxides. Supplementary Fig. 2 shows the composition map where A-site is partially occupied by K. Strontium, an ideally sized cation sitting at A-site can form plenty of stable perovskites, supporting diverse B-site substitutions for tailoring the properties of materials. Alkaline earth metals in A-site generally show high perovskite likelihood. Likewise, these elements can replace toxic lead in many useful perovskites.
To assess how well this fingerprint proximity oriented approach contrasts with supervised parametric models, we evaluated the crystal symmetry detection performance of GBC, DT, RF and SVM classifiers using LOOCV for unbiased comparison. The feature space is chosen to be the same as used in perovskites classification step. The performance metrics are summarized in Table 2. GBC achieves the highest crystal system classification accuracy of 78.8% (see Fig. 4d) while the proposed scheme, relying only on the 2D fingerprints, outperforms supervised algorithms such as SVM and RF. The trend is similar for space group classification. Most importantly, unsupervised deep learning techniques are quite rigorous when finding their own representations, and advantageous in inversely exploring the compositional space, which is not feasible in supervised ML regime. For instance, when the database is of sufficient size and diversity, the representations learned by VAE form an information-embedded topology that can be used not only to predict the crystal structure of unknown compositions but also to discover compositions analogous to target experimental materials, a point that we come back later in this paper.