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    1088

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    The Phonopy code is a helpful resource to obtain vibration related quantities such as phonon band structure and density of states, dynamic structure factor, and Grüneisen parameters [ 86 ].

    In summary, DFT is a mature theory which is currently the undisputed choice of method for electronic structure calculations.

    A number of papers and reviews are presented in the literature [ 87 — 92 ], facilitating the widespread of the theory and, thus, the entry of researchers into the field of computational solid state physics, materials science, and quantum chemistry.

    Although the implementations of DFT take place in many codes and scopes see table 1 , it has been shown recently that the results are consistent as a whole [ 34 ].

    Table 1. Selection of DFT codes according to their basis types. DFT calculations provide a reliable method to study materials once the crystalline or molecular structure is known.

    Based on the Hellman—Feynman theorem [ ], one can use DFT calculations to find a local structural minima of materials and molecules. However, a global optimization of such systems is a much more involved process.

    The possible number of structures for a system containing N atoms inside a box of volume V is huge, given by the combinatorial expression.

    This is a global optimization problem in a high-dimensional space, which has been tackled by several authors. Here we discuss two of the most popular methods proposed in the literature, namely evolutionary algorithms and basin hopping optimization.

    Owing to the fact that not all configurations in this landscape are physically acceptable i. One way of achieving such restriction is by means of evolutionary algorithms, where the survival of the fittest candidate structures is taken into account, thus restricting the search to a small region of the configurational space.

    Introducing mating operations between pairs of candidate structures and mutation operators on single samples, a series of generations of candidate structures is created, and in each of these series only the fittest candidates survive.

    The search is optimized by allowing local relaxation, via DFT or molecular dynamics MD calculations, of the candidate structures, thus avoiding nonphysical configurations, such as too short bond lengths.

    Evolutionary algorithms have been used to find new materials, such as a new high-pressure phase of Na [ — ].

    Another popular method of theoretical structure prediction is basin hopping [ , ]. In this approach, the optimization starts with a random structure that is deformed randomly given a threshold, which is in turn brought to an energy minima, via e.

    DFT calculations. If the reached minima are distinct from the previous configuration, the Metropolis criterion [ ] is used to decide if the move is accepted or not.

    If the answer is yes, it is said that the system hopped between neighboring basins. Owing to the fact that distinct basins represent distinct local structural minima, this algorithm probes the configurational space in an efficient way.

    Other methods of global optimization and theoretical structure prediction of molecules and materials comprise random structure searching AIRSS [ ], particle-swarm optimization methods [ , ], parallel tempering, minima hopping [ ], and simulated annealing.

    The so-called Inverse Design , is an inversion of the traditional direct approach , discussed in section 1. Strategies for direct design usually fall into three categories: descriptive, which in general interpret or confirm experimental evidence; predictive, which predicts novel materials or properties; or predictive for a material class, which predicts novel functionalities by sweeping the candidate compound space.

    The inverse mapping, from target properties to the material was proposed by Zunger [ ] as a means to drive materials discovery presenting specific functionalities.

    According to his inverse design framework, one could find the desired property in known materials, as well as discover new materials while searching for the functionality.

    This can be seen as another global optimization task, but instead of finding the minimum energy structure, it searches for the structure that maximizes the target functionality figure of merit.

    This can be done in three ways: i search for a global minimum using local optimization methods, e. A number of examples have been reported as a successful application of inverse design principles, such as the discovery of non-toxic, high efficient halide perovskites solar absorbers [ ].

    As discussed in section 2 , great advances in simulation methods occurred in the last decades. At the same time, even greater evolution was observed in computational science and technologies.

    Therefore, as time progresses the computational capacity is rapidly increasing. This results in a major reduction in the time used to perform calculations, so a relatively larger time is spent on simulations setup and analysis.

    This changed the theoretical workflow and led to new research strategies. Instead of performing many manually-prepared simulations, one can now automate the input creation and perform several even millions simulations in parallel or sequentially.

    This development is presented in figure 6 and the approach is called high-throughput [ ]. Figure 6. Time spent for calculations and similarly for experiments as a function of technological developments.

    With the computer technological advances, the calculation step can be less time consuming than the setup construction and the results analysis.

    Adapted from [ ]. The idea is to generate and store large quantities of thermodynamic and electronic properties by means of either simulations or experiments for both existing and hypothetical materials, and then perform the discovery or selection of materials with desired properties from these databases [ 13 ].

    This approach does not necessarily involve ML, however, there is an increasing tendency to combine these two methodologies in materials science, as already shown in figure 1.

    Importantly, the HT approach is compatible with theoretical, computational, and experimental methodologies.

    The main hindrance of a given method is the time necessary to perform a single calculation or measurement. The HT engine has to be fast and accurate in order to produce massive amounts of data in a reasonable time, otherwise, its purpose is lost.

    Despite the HT generality, here we are mainly interested in its use in the context of first principles DFT calculations and its adapted strategies, discussed in section 2.

    The implementation of HT-DFT methods is usually performed in three main steps: i thermodynamic or electronic structure calculations for a large number of synthesized and hypothetical materials; ii systematic information storage in databases and; iii materials characterization and selection: data analysis to select novel materials or extract new physical insight [ 13 ].

    The great interest in the use of this methodology, the strong diffusion of methods and algorithms for data processing, and the wide acceptance of ML as a new paradigm of science, have resulted in intensive implementation work to create codes to manage calculations and simulations, as well as materials repositories that allow sharing and distributing results obtained in these simulations, i.

    In general, this is performed in high-performance computers HPC with multi-level parallel architectures managing hundreds of simulations at once.

    A principled way for database construction and dissemination related to step ii is the FAIR concept, which stands for findable, accessible, interoperable, and reusable [ , ].

    Meanwhile, item iii usually referred to as materials screening or high-throughput virtual screening, is performed via filtering the properties provided by the materials repositories.

    In a certain way, this could represent a difficulty, since the information provided by the repositories does not necessarily contain the properties of interest, requiring that each research group perform their own HT calculations, which in many cases results in updates of the databases.

    Thus, in recent years, there has been a considerable increase of materials databases. In table 2 the most used HT theoretical and experimental databases are presented with a brief description.

    Table 2. High-Throughput databases, codes, and tools according to source and purpose. We define a complete package for HT as a multi-engine code that can generate, manipulate, manage and analyze the simulation results.

    On the other hand, the profusion of experimental materials databases is less diverse. The main difference between the two databases is the inclusion of organic, metal-organic compounds and minerals in the COD database.

    Despite the complexities involved in steps i and ii , the third step is more significant. In iii the researcher inquiries the database in order to discover novel materials with a given property, to gain insight on how to modify an existent one, or to extract a subset of materials for further investigations, which involves more calculations or not.

    The quality of the inquiry will determine the success of the search. This is usually performed via a constraint filter or a descriptor, which will be used to separate the materials with the desired property, or a proxy variable.

    We extend the discussion of this process in the next section. Materials screening or mining can be seen as an integral part of a HT workflow, but here we highlight it as a step on its own.

    In a rigorous definition, HT concerns the high-volume data generation step, whereas screening or mining process refers to the application of constraints to the database in order to filter or select the best candidates according to the desired attributes.

    The database is generally screened in sequence through a funnel-like approach, where materials satisfying each constraint pass to the next step, while those who fail to meet one or more of them are eliminated [ 21 ].

    A final step may be to evaluate what characteristics make the top candidates perform best in the desired property, and then predict if these features can be improved further.

    Thus, every material who satisfied the various criteria can be optionally ranked according to a problem-defined merit figure, and then this subgroup of selected materials can be additionally investigated or used in applications.

    The constraints can be descriptors derived from ML processes or filters guided by the previous understanding of the phenomena and properties, or even guided by human intuition.

    Traditionally, descriptors construction requires an intimate knowledge of the problem. The descriptor can be as simple as the free energy of hydrogen adsorbed on a surface, which is a reasonable predictor of good metal alloys for hydrogen catalysis [ ].

    Or more complex such as the variational ratio of spin-orbit distortion versus non-spin-orbit derivative strain, which was used to predict new topological insulators using the AFLOWLIB database [ ].

    Although materials screening procedure has as its final objective the materials prediction and selection, more complex properties, e. Specifically, the filters used for the screening can be descriptors obtained via ML techniques.

    In the same way, the ML process can, in turn, depend on an initial selection of materials. This initial step is to restrict the data set exclusively to materials that potentially exhibit the property of interest.

    For example, in the prediction of topological insulators protected by the time-reversal symmetry, compounds featuring a non-zero magnetic moment are excluded from the database, as we discuss in section 3.

    In figure 7 , the materials screening process is schematically presented. As discussed, the first step consists in defining the design principles, i.

    Subsequently, these filters are used following a funnel procedure. In the ideal scenario, the filters must be applied in a hierarchical way if possible, since this could give information about the mechanisms behind the materials properties.

    Finally, the materials must be organized according to their performance, i. After passing through the filters, if there are candidates that satisfy the criteria, a set of selected materials will be obtained, which could lead to novel technological or scientific applications.

    Figure 7. The materials screening process as a systematic materials selection strategy based on constraints filters. Having presented the most used approaches used to generate large volumes of data, now we examine the next step of dealing and extracting knowledge from the information obtained.

    Exploring the evolution of the fourth paradigm of science, a parallel can be made between the Wigner's paper 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences' [ ] to the nowadays 'The Unreasonable Effectiveness of Data' [ ].

    What makes this unreasonable effectiveness of data in recent times? A case can be made for the fifth 'V' of big data figure 3 : extracting value from the large quantity of data accumulated.

    How is this accomplished? Through machine learning techniques which can identify relationships in the data, however complex they might be, even for arbitrarily high-dimensional spaces, inaccessible for human reasoning.

    ML can be defined as a class of methods for automated data analysis, which are capable of detecting patterns in data.

    These extracted patterns can be used to predict unknown data or to assist in decision-making processes under uncertainty [ ].

    The traditional definition states that the machine learning, i. This research field evolved from the broader area of artificial intelligence AI , inspired by the s developments in statistics, computer science and technology, and neuroscience.

    Figure 8 shows the hierarchical relationship between the broader AI area and ML. Figure 8. Hierarchical description and techniques examples of artificial intelligence and its machine learning and deep learning sub-fields.

    Much of the learning algorithms developed have been applied in areas as diverse as finances, navigation control and locomotion, speech processing, game playing, computer vision, personality profiling, bioinformatics, and many others.

    In contrast, an AI loose definition is any technique that enables computers to mimic human intelligence. This can be achieved not only by ML, but also by 'less intelligent' rigid strategies such as decision trees, if-then rules, knowledge bases, and computer logic.

    Recently, an ML subfield that is increasingly gaining attention due to its successes in several areas is deep learning DL [ ]. It is a kind of representation learning loosely inspired by biological neural networks, having multiple layers between its input and output layers.

    A closely related field and very important component of ML is the source of data that will allow the algorithms to learn from. This is the field of data science, which we introduced in section 1.

    The set X is named feature space and an element x from it is called a feature or attribute vector , or simply an input. With the learned approximate function , the model can then predict the output for unknown examples outside the training data, and its ability to do so is called generalization of the model.

    There are a few categories of ML problems based on the types of inputs and outputs handled, the two main ones are supervised and unsupervised learning.

    In unsupervised learning , also known as descriptive, the goal is to find structure in the data given only unlabeled inputs , in which the output is unknown.

    If f X is finite, the learning is called clustering , which groups data in a known or unknown number of clusters by the similarity in its features.

    On the other hand, if f X is in , the learning is called density estimation , which learns the features marginal distribution.

    Another important type of unsupervised learning is dimensionality reduction , which compresses the number of input variables for representing the data, useful when f X has high dimensionality and therefore a complex data structure to detect patterns.

    If the output y i type is a categorical or nominal finite set for example, metal or insulator , it is called a classification problem, which predicts the class label for unknown samples.

    Else, if the outputs are continuous real-valued scalars , it is then called a regression problem, which will predict the output values for the unknown examples.

    These types of problems and their related algorithms which we introduce in section 2. Figure 9. Machine learning algorithms and usage diagram, divided into the main types of problems: unsupervised dimensionality reduction and clustering and supervised classification and regression learning.

    All Rights Reserved. Used with permission. A typical ML workflow can be summarized as follows [ ]:. In the present context of materials science, we explore the steps: i data collection in sections 2.

    Thus, the task of constructing such an algorithm is a case-by-case study. Such dataset can be of two types: either labeled or unlabeled. In the first case, the task at hand is to find the mapping between data points and corresponding labels by means of a supervised learning algorithm.

    On the other hand, if no labels are present in the dataset, the task is to find a structure within the data, and unsupervised learning takes place.

    Owing to the large abundance of data, one can easily obtain feature vectors of overwhelmingly large size, leading to what is referred to as 'the curse of dimensionality'.

    In this case, the matrix containing these number is flattened into an array of length n 2 which is the feature vector, describing a point in a high dimensional space.

    Due to the exponential dependency, a huge number of dimensions is easily reachable for average sized images. Memory or processing power become limiting factors in this scenario.

    A key point is that within the high-dimensional data cloud spanned by the dataset, one might find a lower dimensional structure.

    The set of points can be projected into a hyperplane or manifold, reducing its dimensionality while preserving most of the information contained in the original data cloud.

    A number of procedures with that aim, such as principal component analysis PCA in conjunction with single value decomposition SVD are routinely employed in ML algorithms [ ].

    In a few words, PCA is a rotation of each axis of the coordinate system of the space where the data points reside, leading to the maximization of the variance along these axes.

    The way to find out where the new axis should point to is by obtaining the eigenvector corresponding to the largest eigenvalue of the X T X , where X is the data matrix.

    Once the largest variance eigenvector, also referred to as the principal component, is found, data points are projected into it, resulting in a compression of the data, as is depicted in figure Figure Principal component analysis PCA performed over a 3D dataset with 3 labels given by the color code left resulting in a 2D dataset right.

    A variety of ML methods is available for unsupervised learning. One of the most popular methods is k -means [ ], which is widely used to find classes within the dataset.

    Once the number of centroids k is chosen and their starting position is selected , e. First, the distances of the data points to each centroid are calculated, and the points are labeled y i as belonging to the subgroup corresponding to the closest centroid.

    Next, a new set of centroids is computed by averaging the positions of the class members of each group. The two steps are described by equations 12 and 13 ,.

    Convergence is reached when no change in the assigned labels is observed. The choice of the starting positions for the centroids is a source of problems in k -means clustering, leading to different final clusters depending on the initial configuration.

    A common practice is to run the clustering algorithm several times and consider the final configuration as the most representative clustering.

    Hierarchical Clustering is another method employed in unsupervised learning which can be found in two flavors, either agglomerative or divisive.

    The former can be described by a simple algorithm: one starts with n classes, or clusters, one containing a single example from the training set, and then measures the dissimilarity d A , B between pairs of clusters labeled A and B.

    The two clusters with the smallest dissimilarity, i. The process is then repeated recursively until only one cluster, containing all the training set elements, remains.

    The process can be better visualized by plotting a dendrogram, shown in figure There is certain freedom into choosing the measure of dissimilarity d A , B , and three main measures are popular.

    First, the single linkage takes into account the closest pair of cluster members,. Second, complete linkage considers the furthest or most dissimilar pair of each cluster,.

    The particular form of d ij can also be chosen, usually being considered the Euclidean distance for numerical data. Unless the data at hand is highly clustered, the choice of the dissimilarity measure can result in distinct dendrograms, and thus, distinct clusters.

    As the name suggests, divisive clustering performs the opposite operation, starting from a single cluster containing all examples from the data set and divides it recursively in a way that cluster dissimilarity is maximized.

    Similarly, it requires the user to determine the cut line in order to cluster the data. In the case where not only the features X but also the labels y i are present in the dataset, one is faced with a supervised learning task.

    Within this scenario, if the labels are continuous variables, the most used learning algorithm is known as Linear Regression.

    It is a regression method capable of learning the continuous mapping between the data points and the labels. Its basic assumption is that the data points are normally distributed with respect to a fitted expression,.

    Once the ML model is considered trained, its performance can be assessed by a test set, which consists of a smaller sample in comparison to the train set that is not used during training.

    Two main problems might arise then: i if the descriptor vectors present an insufficient number of features, i. Roughly speaking, these are the two extremes of model complexity, which is in turn directly related to the number of parameters of the ML model, as is depicted in figure The optimum model complexity is evaluated against the prediction error given by the test set.

    Adapted with permission from [ ]. One is not restrained to choose a specific metric for the regularization term in equation 20 : methods for interpolation, such as elastic net [ , ], are capable of finding an optimal combination of regularization parameters.

    Another class of supervised learning, known as classification algorithms, is broadly used when the dataset is labeled by discrete labels.

    A very popular algorithm for classification is logistic regression, which can be interpreted as a mapping of the predictions made by linear regression into the [0, 1] interval.

    The desired binary prediction can be obtained from. As an example, the sigmoid function along with some prediction from a fictitious dataset is presented in figure Usually one considers that data point x i belongs to class labeled by y i if , even though the predicted label can be interpreted as a probability.

    The gray arrow points out to the incorrectly classified points in the dataset. The data point labels correspond to the distinct colors of the scatter points while the assignment to each cluster, defined by their centroids black crosses , corresponds to the color patches.

    Horizontal lines denote the merging process of two clusters. The number of cuts between a horizontal line and the cluster lines denotes the number of clusters at a given height, which in the case of the gray dashed line is five.

    In the case of classification, the cost function is obtained from the negative log-likelihood. Notice that logistic regression can also be used when the data presents multiple classes.

    In this case, one should employ the one-versus-all strategy, which consists on training n logistic regression models, one for each class, and predicting the labels using the classifier that presents the highest probability.

    By proposing a series of changes in the logistic regression, Cortes and Vapnik introduced one of the most popular ML classification algorithms, support vector machines SVMs [ ].

    Such changes can be summarized by the introduction of the following cost function,. Insertion of max z , 0 into the cost function leads to a maximization of a classification gap containing the decision boundary in the data space.

    The optimization problem described above can also be interpreted as the minimization of subject to the constraints for all belonging to the training set.

    In fact, by writing the Lagrangian for this constrained minimization problem, one ends up with an expression that corresponds to the cost function given by equation One of the most powerful features of SVMs is the kernel trick.

    This makes possible to express the decision rule as a function of dot products between data vectors. The kernel trick consists into transforming the vectors in the dot products using a mapping that takes the data points into a larger dimensional space, where a decision boundary can be envisaged.

    Moreover, any transformation that maps the dot product into a vector-pair function has been proven to work similarly to what was described above.

    A couple of the most popular kernels are the polynomial kernel, , and the Gaussian kernel, also known as radial basis function RBF kernel,.

    The Gaussian kernel usage is usually interpreted as a pattern-matching process, by measuring the similarity between data points in high-dimensional space.

    Up to this point, all classification algorithms presented are based on discriminative models , where the task is to model the probability of a label given the data points or features.

    Another class of algorithm capable of performing the same task, but using a different approach of a generative model , where one aims to learn the probability of the features given the label can be derived from the famous Bayes formula for calculation of a posterior probability,.

    Its assumption enables one to rewrite the posterior probability from equation 26 as. Usually the denominator in this equation is disregarded since it is a constant for all possible values of y , and the probability is renormalized.

    The training step for this classifier comprises the tabulation of the priors p y for all labels in the training set as well as the conditional probabilities from the same source.

    Another popular and simple classification algorithm is k -nearest neighbors kNN. Based on similarity by distance, this algorithm does not require a training step, which makes it attractive for quick tasks.

    In short, given a training set composed of data points in a d -dimensional space , kNN calculates the distance between these points and an unseen data point x ,.

    Once all distances are obtained, the class of x is simply the class of the majority of its k nearest neighbors. If there is no majority, its class is assigned randomly from the most frequent labels of the neighbors.

    On the other hand, a regressor based on kNN is obtained by averaging the continuous label values of the nearest neighbors.

    As mentioned earlier for other ML algorithms, the value of k cannot be learned in this case, leaving the task of choosing a sensitive k to the user.

    For classification tasks, different choices of such hyperparameter might result in distinct partitionings of the data cloud, which can be visualized as the Voronoi tessellation diagrams in figure Finally, some ML algorithms are suited both for classification and regression.

    Decision Trees are a popular and fast ML algorithm that can be used in both cases. Since it can be implemented in a variety of flavors, we chose to explain briefly the workings of two of the most popular implementations, the classification and regression trees, or CART, and the C4.

    Both methods are based on the partitioning of the data space, i. Each node of the tree contains a question which defines such a partition.

    When no further partitioning of the space is possible, each disjoint subspace, referred to as the leaves, contains the data points one wishes to classify or predict.

    This is done in such a way to maximize the ratio between information gain and potential information that can be obtained from a particular partitioning or test B.

    The potential information P S , B that such partitioning can provide is given by. Partitioning takes place up to the point where the nodes contain only examples of one class or examples of distinct classes that cannot be distinguished by their attributes.

    On the other hand, CART is a decision tree method which is capable of binary partitioning only. In the case of classification tasks, it uses a criterion for splitting which is based on the minimization of the Gini impurity coefficient.

    If one is interested in using CART for a regression task, there are two main differences to be considered. First, the nodes predict real numbers instead of classes.

    Second, the splitting criterion, in this case, is the minimization of the resubstitution estimate, which is basically a mean squared error.

    The consequence of such partitioning is that for each partition, the predicted value is the average of the values within that partition. Thus, CART outputs a piecewise constant function for regression.

    One of the major issues with regression trees is that once they are trained, most of the time they suffer from overfitting. A couple of strategies to overcome this problem have been proposed, such as pruning the trees' structures in order to increase its generalization power, loosing however some of their accuracies.

    More advanced methods include Random Forests, which is an ensemble method based on training several decision trees and averaging their predictions [ ].

    In this case, the trees are smaller versions of the structures described previously, trained using a randomly chosen subset of the features of the dataset, and usually a bootstrap sample of the same set.

    In some sense, building a series of weaker learners and combining their predictions enables the algorithm to learn particular features of the dataset and better generalize to new, unseen data.

    Artificial Neural Networks ANNs corresponds to a class of algorithms that were, at least in their early stages, inspired by the brain structure.

    An ANN can be described as a directed weighted graph, i. Many kinds of ANNs are used for a variety of tasks, namely regression, and classification, and some of the most popular architectures for such networks are feed-forward, recurrent, and convolutional ANNs.

    The main difference between these architectures is basically on the connection patterns and operations that their neurons perform on data.

    Example of a feed-forward ANN with N hidden layers and a single neuron in the output layer. Red neurons represent sigmoid activated units see equation 35 while yellow ones correspond to the ReLU activation equation Typically in an ANN, an input layer receives the descriptor vectors from the training set, and a series of non-linear operations is performed as data forward propagates through the subsequent hidden layers.

    Finally, the outcome of the processing is collected at the output layers, which can be either a binary or multinary probabilistic classification, or even a continuous mapping as in a linear regression model.

    In an ANN, the input of the i -th neuron in the k -th layer is a function of the outputs of the previous layer.

    The element is referred to as the bias, because it is not part of the linear combination of inputs. The input is then transformed via a non-linear, or activation function, such as the sigmoid,.

    Such intricate structure can be used for regression when the measure of accuracy is the squared error given by equation For a single class classification task, an ANN should output a single sigmoid-activated neuron, corresponding to the probability of the input example belonging to the particular class.

    In this case, the measure of accuracy is the same as in the logistic regression algorithm, the cross-entropy given by equation In case one is interested in multi-class classification, a softmax activation should be used, corresponding to the probability of output vector representing a member of class y i ,.

    Optimal values for the parameters are found by calculating the gradient of L with respect to these parameters and performing gradient descent minimization.

    This process is referred to as back-propagation. In a nutshell, using ANNs for machine learning tasks comprise a series of steps: i random initialization of the weights , ii forward pass training examples and computing their outcomes, iii calculate their deviations from the corresponding labels via the loss function, iv obtain the gradients of that function with respect to the network weights via back-propagation, and finally v adjust the weights in order to minimize the loss function.

    Such process might be performed for each example of the training set at a time, which is called online learning, or using samples of the set at each step, being referred to as mini-batch or simply batch learning.

    A ML supervised learning algorithm is considered trained when its optimal parameters given the training data are found, by minimizing a loss function or negative log likelihood.

    However, the hyperparameters usually cannot be learned in this manner, and the study of the performance of the model over a separate set, referred to as the validation set, as a function of such parameters is of order.

    This process is referred to as validation. The usual way of doing so is separating the dataset into 3 separate sets: the training, validation, and test sets.

    It is expected that their contents are of the same nature, i. The learning process is then performed several times in order to optimize the model.

    Finally, by using the test set, one can confront the predictions with the actual labels and measure how far off the model is performing. The optimal balance is represented in figure When a limited amount of data is available for training, removing a fraction of that set in order to create the test set might impact negatively the training process, and alternative ways should be employed.

    One of the most popular methods in this scenario is k -fold cross-validation, which consists in partitioning the train set in k subsets, and train the model using of the subsets and validate the trained model using the set that was not used for training.

    This process is performed k times and the average of each validation step is used to average the performance,. Other parameters that might not seem so obvious, such as the pruning level of binary trees or the number of features one selects in order to create the ensemble for a random forest can also be optimized in the same way.

    The error is then evaluated for a series of values of the parameters and the value that minimizes the prediction or test error is selected in this case.

    There are many different ways of evaluating the performance. As an example, in binary or multinary classification tasks, the use of confusion matrices, where the number of correctly predicted elements are presented in the diagonal entries while the elements that were incorrectly predicted are counted in the off-diagonal entries, is very common.

    One can think of the vertical index as the actual labels and horizontal index as the predictions, and false F positives P or negatives N are positive predictions for negative cases and the converse, respectively.

    The receiver operating characteristic ROC curve is also routinely used, being the plot of the true T positive rate versus the false positive rate with changing threshold.

    In the case of regression tasks, there are several measures of the fitting accuracy. The mean absolute error , measures deviations in the same unit as the variable, and also is not sensitive to outliers.

    There is the normalized version expressed in percentage. The mean squared error combines bias and variance measurements of the prediction.

    The MSE, i. Finally, the statistical coefficient of determination R 2 is also used, defined as , where the total sum of squares is and the residual sum of squares is.

    Inspired by the success of applied information sciences such as bioinformatics, the application of machine learning and data-driven techniques to materials science developed into a new sub-field called 'Materials Informatics' [ ], which aims to discover the relations between known standard features and materials properties.

    These features are usually restricted to the structure, composition, symmetry, and properties of the constituent elements.

    Recasting the learning problem stated in section 2. Naturally, this question has always been at the heart of materials science, what changes here is the way to solve it.

    Specifically, one has to give a known example dataset to train an approximate ML model and then make predictions on materials of interest that are outside the dataset.

    Ultimately the inverse question can also be answered see section 2. A model must be constructed to predict properties or functional relationships from the data.

    The model is an approximate function that brings the inputs materials features to the outputs properties. As such, it can be seen as a phenomenological or empirical model, because it arrives at a heuristic function that describes the available data.

    The ML process is expected to provide features-property relationships that are hidden to human capacities. In the context of science paradigms discussed in section 1.

    Even though, these approximate models can lead to better understanding and ultimately aid in the construction of theories.

    In Feynman's words: ' We do not know what the rules of the game are; all we are allowed to do is to watch the playing.

    Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we mean by fundamental physics.

    The machine learning task for constructing models for materials is an applied version of the general ML workflow presented in section 2.

    As discussed, the supervised tasks can be divided into two groups: learning of a numerical material property or materials classification.

    In the first case, the ML process aims to find a functional form f x for a numeric target property, requiring the use of methods such as regression.

    Otherwise, classification aims to create 'materials maps', in which compounds or molecules exhibiting different categories of the same property are accordingly identified by class labels.

    For example, magnetic and non-magnetic systems non-zero and zero magnetic moment , or compounds stable at zinc blende or rock salt structures form two different classes.

    In these maps, the overlap between the classes must be zero, as schematically represented for a Voronoi diagram depicting the k-means classification see figure Thus, the class of a material outside the training set can be identified only by its position on the map.

    In section 3 , we discuss examples and progress based on these kinds of material informatics tasks.

    Here, we first outline the usually followed process. The materials informatics workflow consists basically of the same general components see section 2.

    The complete materials informatics workflow is summarized in figure The above steps are essentially incorporating ML techniques to update the historical way for addressing materials science problems.

    Therefore, there are some relevant examples that follow the discussed strategy even before these computational developments. The periodic table of elements is an influential example of a successful representation, i.

    Impressively, this organization leads to a two-dimensional description given by two simple numbers, the table row and column. Only 50 years later, quantum mechanics brings the physical reasoning behind this two-dimensional descriptor, the shell structure of the electrons.

    Despite this delayed interpretation, the periodic table anticipated undiscovered elements and their properties, assuring its predictive power [ ].

    On the other hand, the challenge to sort all materials is much complex, since there are potentially millions of materials instead of only elements.

    Additionally, only a small fraction of these compounds have their basic properties determined [ ]. This problem is even more complex for the infinitely large dataset formed by the all possible combinations of surfaces, interfaces, nanostructures, and organic materials, in which the complexity of materials properties is much higher.

    Therefore, it is reasonable to suppose that materials with promising properties are still to be discovered in almost every field [ ].

    In practice, several software packages and tools for different types of ML tasks exist, and are presented in table 3. General purpose codes work for the various types of problems section 2.

    Materials specific codes aid in the different steps of the MI workflow. These include data curation and representation by transforming general materials information compositional, structural, electronic, etc into feature vectors details in the next section 2.

    Table 3. Selection of materials informatics and machine learning codes and tools. Adapted from [ 22 ]. Finally, we now discuss an essential question regarding ML research: when ML should or not be employed and what kind of problems it tackles.

    An obvious crucial prerequisite is the availability of data, which should be consistent, sufficient, validated, and representative of the behavior of interest to be described.

    Once more we emphasize this requirement and thus, the common data generation process is generally better suited to traditional or HT approaches, at least initially.

    Additionally, one has to consider the strengths of machine learning methods, which can manage high-dimensional spaces in searching for relationships in data.

    The patterns discovered are then explicit encoded, rendering computational models that can be manipulated. In contrast, if human intuition can produce a physical model, ML is probably not needed by the problem.

    Therefore, ML methods are best suited to problems where traditional approaches have difficulties. Although it is not always clear to specify, if a problem can be identified into one of the general ML problem types described in section 2.

    Historically, areas which have questions with these characteristics have had successful applications of ML methods, such as in automation, image and language processing, social, chemical and biological sciences, and in recent times many more examples are emerging.

    Based on these characteristics, we glimpse on the common types of materials science applied problems which make use of data-driven strategies, and that are exemplified in section 3.

    A related strategy is to replace the description of a very complex or expensive property that is somewhat known, at least for a small class of materials by a simpler ML model, rendering its calculation less expensive.

    If properly validated, this model can then predict the complex property for unknown examples, expanding the data set. In the context of materials discovery and design, this strategy can be employed as a form of extending the data set before the screening, where the initial expensive data leads to more data through the ML model, which can then be screened for novel promising candidates.

    Other problems use feature selection techniques to discover approximate models and descriptors, which aid in the phenomenological understanding of the problem.

    Another type of problem and perhaps the most abundant is the clear advantageous problems in which expensive calculations can be replaced by a much more efficient model, such as replacing altogether DFT calculations for ML models such as in obtaining atomistic potentials for MD simulations, predicting the value of different properties gap, formation and total energies, conductivity, magnetization, etc.

    The representation of materials is a crucial component determining the machine learning performance. Only if the necessary variables are sufficiently represented then the learning algorithm will be able to describe the desired relationship.

    A representation objective is to transform materials characteristics such as composition, stoichiometry, structure, and properties into a quantitative numerical list, i.

    These variables used to represent materials characteristics are called features , descriptors , or even fingerprints. A general guideline can be expressed by a variant of Occam's razor which is a paraphrase famously attributed to Einstein, a representation should be 'as simple as possible, but not simpler'.

    For any new ML problem, the feature engineering process is responsible for most of the effort and time used in the project [ ].

    Other helpful characteristics for representations are having a high target similarity similarity between representation and original represented function , and from a computational perspective, functions of fixed dimensionality, and smooth and continuous, which ensures differentiability.

    These requisites presented act to assure that the models will be efficient with only the essential information.

    Such field has shown a relative degree of success, but also inconsistent performance of its models, arising from a lack of either proper domain of applicability, satisfactory descriptors, or machine learning validation [ ].

    Generally, a material can be described in several ways, of increasing complexity degree, depending on each problem needs. The simplest way is using only the chemical features such as atomic element types and stoichiometric information, which involves no structural characterization, therefore being more general but less specific to distinct polymorphs which can present different properties.

    This kind of rough description manages to describe general trends among very different types of materials. In order to increase the description capability of the ML models, higher complexity can be handled by introducing more relevant information available [ ].

    For descriptors based on elemental properties [ , , , ] this involves including and combining elements properties and statistics of these such as the mean, mean absolute deviation, range, minimum, maximum and mode.

    Stoichiometric attributes can include the number of elements, fractions, and norms. Even beyond, ionic character [ , ] and electronic structure attributes [ , , ], fingerprints [ ] and statistics can be included, to account for more intricate relationships.

    Including the structural information of the high dimensional space of atomic configurations [ ] is not a simple task. Common structural representations are not directly applicable to computational descriptions.

    Materials, especially solids, are commonly represented by their Bravais matrix and a basis, including the information of the translation vectors and the atom types and positions, respectively.

    For machine learning purposes, this representation is not suitable due to not being unique. In case of structural input, the requisites presented above indicate that the chemical species and atomic coordinates should suffice for an efficient representation.

    As such, the models should preserve the systems symmetries such as translational, rotational, and permutational. Ultimately, the representation objective is to ensure accuracy comparable to or superior than quantum mechanics calculations, for a wide range of systems, but with reduced computational cost.

    These so-called structural fingerprints are increasingly used to describe the potential energy surfaces PES of different systems, leading to force fields for classical atomistic simulations with QM accuracy, but with computational cost orders of magnitude lower and also linear scaling n behavior with the number of atoms.

    Most of these potentials benefit from chemical locality, i. Notable examples are the Gaussian Approximation Potentials GAPs [ , ], Behler—Parrinello high-dimensional neural network potentials [ , ], and Deep Potential molecular dynamics [ ].

    Related to the structural representation, a scoring parameter to identify dimensionality of materials was recently developed [ ].

    There are also methods for structural similarity measurements improving upon the commonly used root-mean-square distance RMSD [ ], such as fingerprint distances [ , ], functional representation of an atomic configuration FRAC [ ], distance matrix and eigen-subspace projection function EPF [ ], and the regularized entropy match REMatch [ ], used with SOAP.

    There is a vast collection of descriptor proposals in the literature that include more complex representations than the simple elemental properties discussed above, ranging from molecule-oriented fingerprints to descriptors for extended materials systems and tensorial properties.

    We now present a relatively chronological list used in recent materials research, which is considerable but not exhaustive. An important open discussion regards the interpretability [ , ] of the descriptors and consequently of the models obtained with ML [ , ].

    As already stated, one of the materials science objectives is to discover governing relationships for the different materials properties, which enable predictive capacity for a wide materials space.

    A choice can be made when choosing and designing the descriptors to be used. When prediction accuracy is the main goal, ML methods can be used as a black-box, and the descriptor interpretation and dimensionality are secondary.

    On the other hand, if the goal in addition to accuracy is understanding, physically meaningful descriptors can provide insight into the relationship described, and help even to formulate approximate and rough phenomenological models [ ].

    This cycle is presented in figure The revised connectivity between the four science paradigms. From empirical data to fundamental theories, which are realized in computational simulations, generating even more data.

    Statistical learning in turn can obtain simple phenomenological models that aids in theoretical understanding. Robinson, p. Klewitz, p. Maria in Cosmedin, but this is not attested in the documents.

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    Papal elections and conclaves. Papal selection before Papal conclave —, —present. Ubi periculum Aeterni Patris Filius Cum proxime Ingravescentem aetatem Romano Pontifici eligendo Universi Dominici gregis De aliquis mutationibus in normis de electione Romani Pontificis Normas nonnullas Categories : 11th-century elections in Europe Papal elections 11th-century Catholicism.

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    Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version. Bishop of Sabina.

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    Bishop of Porto. Chancellor of the Holy Roman Church. Abbot of the monastery of S.

    1088 1088 Normen mitgestalten Ihr Ansprechpartner. Fördermittel für die Sanierung von sechs Bushaltestellen. Lade Empfehlungen Ausgabedatum Originalsprachen Deutsch Produktinformationen auf dieser Seite:. Das Anzünden des Feuers ist um Verbindungen zu anderen Normen Diese Norm zitiert. Für das leibliche Wohl ist wie immer gesorgt. So können Sie beispielsweise Vampire Diaries Serien Stream Staffel 7 Basis einer vorherigen z. Restkarten für Kleinkunst-Abend. Notwendig Funktional Personalisierung Details anzeigen. The error is then evaluated for Boxing Helena series of values Neue Bachelorette 2019 the parameters and the value that minimizes the prediction or test error is selected in this case. Generally, a material can be described in several ways, of increasing complexity degree, depending on each problem needs. Initial developments of each discipline date to many decades before actual adoption by the community. The 2D materials era was initiated with the graphene Australia Film by Novoselov and Geim [ ]. Figure 7. Florida International University. Such a method basically describes the core electrons and corresponding nuclei in a simplified manner, by means of an effective potential which the valence electrons are subject to. Nur Mit Dir Ganzer Film Deutsch such, the models should Breitengrad New York the systems symmetries such as translational, rotational, and permutational.

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    The two steps are described by equations 12 and 13 ,. Convergence is reached when no change in the assigned labels is observed. The choice of the starting positions for the centroids is a source of problems in k -means clustering, leading to different final clusters depending on the initial configuration.

    A common practice is to run the clustering algorithm several times and consider the final configuration as the most representative clustering. Hierarchical Clustering is another method employed in unsupervised learning which can be found in two flavors, either agglomerative or divisive.

    The former can be described by a simple algorithm: one starts with n classes, or clusters, one containing a single example from the training set, and then measures the dissimilarity d A , B between pairs of clusters labeled A and B.

    The two clusters with the smallest dissimilarity, i. The process is then repeated recursively until only one cluster, containing all the training set elements, remains.

    The process can be better visualized by plotting a dendrogram, shown in figure There is certain freedom into choosing the measure of dissimilarity d A , B , and three main measures are popular.

    First, the single linkage takes into account the closest pair of cluster members,. Second, complete linkage considers the furthest or most dissimilar pair of each cluster,.

    The particular form of d ij can also be chosen, usually being considered the Euclidean distance for numerical data.

    Unless the data at hand is highly clustered, the choice of the dissimilarity measure can result in distinct dendrograms, and thus, distinct clusters.

    As the name suggests, divisive clustering performs the opposite operation, starting from a single cluster containing all examples from the data set and divides it recursively in a way that cluster dissimilarity is maximized.

    Similarly, it requires the user to determine the cut line in order to cluster the data. In the case where not only the features X but also the labels y i are present in the dataset, one is faced with a supervised learning task.

    Within this scenario, if the labels are continuous variables, the most used learning algorithm is known as Linear Regression.

    It is a regression method capable of learning the continuous mapping between the data points and the labels. Its basic assumption is that the data points are normally distributed with respect to a fitted expression,.

    Once the ML model is considered trained, its performance can be assessed by a test set, which consists of a smaller sample in comparison to the train set that is not used during training.

    Two main problems might arise then: i if the descriptor vectors present an insufficient number of features, i. Roughly speaking, these are the two extremes of model complexity, which is in turn directly related to the number of parameters of the ML model, as is depicted in figure The optimum model complexity is evaluated against the prediction error given by the test set.

    Adapted with permission from [ ]. One is not restrained to choose a specific metric for the regularization term in equation 20 : methods for interpolation, such as elastic net [ , ], are capable of finding an optimal combination of regularization parameters.

    Another class of supervised learning, known as classification algorithms, is broadly used when the dataset is labeled by discrete labels.

    A very popular algorithm for classification is logistic regression, which can be interpreted as a mapping of the predictions made by linear regression into the [0, 1] interval.

    The desired binary prediction can be obtained from. As an example, the sigmoid function along with some prediction from a fictitious dataset is presented in figure Usually one considers that data point x i belongs to class labeled by y i if , even though the predicted label can be interpreted as a probability.

    The gray arrow points out to the incorrectly classified points in the dataset. The data point labels correspond to the distinct colors of the scatter points while the assignment to each cluster, defined by their centroids black crosses , corresponds to the color patches.

    Horizontal lines denote the merging process of two clusters. The number of cuts between a horizontal line and the cluster lines denotes the number of clusters at a given height, which in the case of the gray dashed line is five.

    In the case of classification, the cost function is obtained from the negative log-likelihood. Notice that logistic regression can also be used when the data presents multiple classes.

    In this case, one should employ the one-versus-all strategy, which consists on training n logistic regression models, one for each class, and predicting the labels using the classifier that presents the highest probability.

    By proposing a series of changes in the logistic regression, Cortes and Vapnik introduced one of the most popular ML classification algorithms, support vector machines SVMs [ ].

    Such changes can be summarized by the introduction of the following cost function,. Insertion of max z , 0 into the cost function leads to a maximization of a classification gap containing the decision boundary in the data space.

    The optimization problem described above can also be interpreted as the minimization of subject to the constraints for all belonging to the training set.

    In fact, by writing the Lagrangian for this constrained minimization problem, one ends up with an expression that corresponds to the cost function given by equation One of the most powerful features of SVMs is the kernel trick.

    This makes possible to express the decision rule as a function of dot products between data vectors. The kernel trick consists into transforming the vectors in the dot products using a mapping that takes the data points into a larger dimensional space, where a decision boundary can be envisaged.

    Moreover, any transformation that maps the dot product into a vector-pair function has been proven to work similarly to what was described above.

    A couple of the most popular kernels are the polynomial kernel, , and the Gaussian kernel, also known as radial basis function RBF kernel,.

    The Gaussian kernel usage is usually interpreted as a pattern-matching process, by measuring the similarity between data points in high-dimensional space.

    Up to this point, all classification algorithms presented are based on discriminative models , where the task is to model the probability of a label given the data points or features.

    Another class of algorithm capable of performing the same task, but using a different approach of a generative model , where one aims to learn the probability of the features given the label can be derived from the famous Bayes formula for calculation of a posterior probability,.

    Its assumption enables one to rewrite the posterior probability from equation 26 as. Usually the denominator in this equation is disregarded since it is a constant for all possible values of y , and the probability is renormalized.

    The training step for this classifier comprises the tabulation of the priors p y for all labels in the training set as well as the conditional probabilities from the same source.

    Another popular and simple classification algorithm is k -nearest neighbors kNN. Based on similarity by distance, this algorithm does not require a training step, which makes it attractive for quick tasks.

    In short, given a training set composed of data points in a d -dimensional space , kNN calculates the distance between these points and an unseen data point x ,.

    Once all distances are obtained, the class of x is simply the class of the majority of its k nearest neighbors. If there is no majority, its class is assigned randomly from the most frequent labels of the neighbors.

    On the other hand, a regressor based on kNN is obtained by averaging the continuous label values of the nearest neighbors.

    As mentioned earlier for other ML algorithms, the value of k cannot be learned in this case, leaving the task of choosing a sensitive k to the user.

    For classification tasks, different choices of such hyperparameter might result in distinct partitionings of the data cloud, which can be visualized as the Voronoi tessellation diagrams in figure Finally, some ML algorithms are suited both for classification and regression.

    Decision Trees are a popular and fast ML algorithm that can be used in both cases. Since it can be implemented in a variety of flavors, we chose to explain briefly the workings of two of the most popular implementations, the classification and regression trees, or CART, and the C4.

    Both methods are based on the partitioning of the data space, i. Each node of the tree contains a question which defines such a partition.

    When no further partitioning of the space is possible, each disjoint subspace, referred to as the leaves, contains the data points one wishes to classify or predict.

    This is done in such a way to maximize the ratio between information gain and potential information that can be obtained from a particular partitioning or test B.

    The potential information P S , B that such partitioning can provide is given by. Partitioning takes place up to the point where the nodes contain only examples of one class or examples of distinct classes that cannot be distinguished by their attributes.

    On the other hand, CART is a decision tree method which is capable of binary partitioning only. In the case of classification tasks, it uses a criterion for splitting which is based on the minimization of the Gini impurity coefficient.

    If one is interested in using CART for a regression task, there are two main differences to be considered. First, the nodes predict real numbers instead of classes.

    Second, the splitting criterion, in this case, is the minimization of the resubstitution estimate, which is basically a mean squared error. The consequence of such partitioning is that for each partition, the predicted value is the average of the values within that partition.

    Thus, CART outputs a piecewise constant function for regression. One of the major issues with regression trees is that once they are trained, most of the time they suffer from overfitting.

    A couple of strategies to overcome this problem have been proposed, such as pruning the trees' structures in order to increase its generalization power, loosing however some of their accuracies.

    More advanced methods include Random Forests, which is an ensemble method based on training several decision trees and averaging their predictions [ ].

    In this case, the trees are smaller versions of the structures described previously, trained using a randomly chosen subset of the features of the dataset, and usually a bootstrap sample of the same set.

    In some sense, building a series of weaker learners and combining their predictions enables the algorithm to learn particular features of the dataset and better generalize to new, unseen data.

    Artificial Neural Networks ANNs corresponds to a class of algorithms that were, at least in their early stages, inspired by the brain structure.

    An ANN can be described as a directed weighted graph, i. Many kinds of ANNs are used for a variety of tasks, namely regression, and classification, and some of the most popular architectures for such networks are feed-forward, recurrent, and convolutional ANNs.

    The main difference between these architectures is basically on the connection patterns and operations that their neurons perform on data. Example of a feed-forward ANN with N hidden layers and a single neuron in the output layer.

    Red neurons represent sigmoid activated units see equation 35 while yellow ones correspond to the ReLU activation equation Typically in an ANN, an input layer receives the descriptor vectors from the training set, and a series of non-linear operations is performed as data forward propagates through the subsequent hidden layers.

    Finally, the outcome of the processing is collected at the output layers, which can be either a binary or multinary probabilistic classification, or even a continuous mapping as in a linear regression model.

    In an ANN, the input of the i -th neuron in the k -th layer is a function of the outputs of the previous layer. The element is referred to as the bias, because it is not part of the linear combination of inputs.

    The input is then transformed via a non-linear, or activation function, such as the sigmoid,. Such intricate structure can be used for regression when the measure of accuracy is the squared error given by equation For a single class classification task, an ANN should output a single sigmoid-activated neuron, corresponding to the probability of the input example belonging to the particular class.

    In this case, the measure of accuracy is the same as in the logistic regression algorithm, the cross-entropy given by equation In case one is interested in multi-class classification, a softmax activation should be used, corresponding to the probability of output vector representing a member of class y i ,.

    Optimal values for the parameters are found by calculating the gradient of L with respect to these parameters and performing gradient descent minimization.

    This process is referred to as back-propagation. In a nutshell, using ANNs for machine learning tasks comprise a series of steps: i random initialization of the weights , ii forward pass training examples and computing their outcomes, iii calculate their deviations from the corresponding labels via the loss function, iv obtain the gradients of that function with respect to the network weights via back-propagation, and finally v adjust the weights in order to minimize the loss function.

    Such process might be performed for each example of the training set at a time, which is called online learning, or using samples of the set at each step, being referred to as mini-batch or simply batch learning.

    A ML supervised learning algorithm is considered trained when its optimal parameters given the training data are found, by minimizing a loss function or negative log likelihood.

    However, the hyperparameters usually cannot be learned in this manner, and the study of the performance of the model over a separate set, referred to as the validation set, as a function of such parameters is of order.

    This process is referred to as validation. The usual way of doing so is separating the dataset into 3 separate sets: the training, validation, and test sets.

    It is expected that their contents are of the same nature, i. The learning process is then performed several times in order to optimize the model.

    Finally, by using the test set, one can confront the predictions with the actual labels and measure how far off the model is performing.

    The optimal balance is represented in figure When a limited amount of data is available for training, removing a fraction of that set in order to create the test set might impact negatively the training process, and alternative ways should be employed.

    One of the most popular methods in this scenario is k -fold cross-validation, which consists in partitioning the train set in k subsets, and train the model using of the subsets and validate the trained model using the set that was not used for training.

    This process is performed k times and the average of each validation step is used to average the performance,. Other parameters that might not seem so obvious, such as the pruning level of binary trees or the number of features one selects in order to create the ensemble for a random forest can also be optimized in the same way.

    The error is then evaluated for a series of values of the parameters and the value that minimizes the prediction or test error is selected in this case.

    There are many different ways of evaluating the performance. As an example, in binary or multinary classification tasks, the use of confusion matrices, where the number of correctly predicted elements are presented in the diagonal entries while the elements that were incorrectly predicted are counted in the off-diagonal entries, is very common.

    One can think of the vertical index as the actual labels and horizontal index as the predictions, and false F positives P or negatives N are positive predictions for negative cases and the converse, respectively.

    The receiver operating characteristic ROC curve is also routinely used, being the plot of the true T positive rate versus the false positive rate with changing threshold.

    In the case of regression tasks, there are several measures of the fitting accuracy. The mean absolute error , measures deviations in the same unit as the variable, and also is not sensitive to outliers.

    There is the normalized version expressed in percentage. The mean squared error combines bias and variance measurements of the prediction.

    The MSE, i. Finally, the statistical coefficient of determination R 2 is also used, defined as , where the total sum of squares is and the residual sum of squares is.

    Inspired by the success of applied information sciences such as bioinformatics, the application of machine learning and data-driven techniques to materials science developed into a new sub-field called 'Materials Informatics' [ ], which aims to discover the relations between known standard features and materials properties.

    These features are usually restricted to the structure, composition, symmetry, and properties of the constituent elements. Recasting the learning problem stated in section 2.

    Naturally, this question has always been at the heart of materials science, what changes here is the way to solve it. Specifically, one has to give a known example dataset to train an approximate ML model and then make predictions on materials of interest that are outside the dataset.

    Ultimately the inverse question can also be answered see section 2. A model must be constructed to predict properties or functional relationships from the data.

    The model is an approximate function that brings the inputs materials features to the outputs properties. As such, it can be seen as a phenomenological or empirical model, because it arrives at a heuristic function that describes the available data.

    The ML process is expected to provide features-property relationships that are hidden to human capacities. In the context of science paradigms discussed in section 1.

    Even though, these approximate models can lead to better understanding and ultimately aid in the construction of theories. In Feynman's words: ' We do not know what the rules of the game are; all we are allowed to do is to watch the playing.

    Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we mean by fundamental physics.

    The machine learning task for constructing models for materials is an applied version of the general ML workflow presented in section 2.

    As discussed, the supervised tasks can be divided into two groups: learning of a numerical material property or materials classification.

    In the first case, the ML process aims to find a functional form f x for a numeric target property, requiring the use of methods such as regression.

    Otherwise, classification aims to create 'materials maps', in which compounds or molecules exhibiting different categories of the same property are accordingly identified by class labels.

    For example, magnetic and non-magnetic systems non-zero and zero magnetic moment , or compounds stable at zinc blende or rock salt structures form two different classes.

    In these maps, the overlap between the classes must be zero, as schematically represented for a Voronoi diagram depicting the k-means classification see figure Thus, the class of a material outside the training set can be identified only by its position on the map.

    In section 3 , we discuss examples and progress based on these kinds of material informatics tasks. Here, we first outline the usually followed process.

    The materials informatics workflow consists basically of the same general components see section 2. The complete materials informatics workflow is summarized in figure The above steps are essentially incorporating ML techniques to update the historical way for addressing materials science problems.

    Therefore, there are some relevant examples that follow the discussed strategy even before these computational developments.

    The periodic table of elements is an influential example of a successful representation, i. Impressively, this organization leads to a two-dimensional description given by two simple numbers, the table row and column.

    Only 50 years later, quantum mechanics brings the physical reasoning behind this two-dimensional descriptor, the shell structure of the electrons.

    Despite this delayed interpretation, the periodic table anticipated undiscovered elements and their properties, assuring its predictive power [ ].

    On the other hand, the challenge to sort all materials is much complex, since there are potentially millions of materials instead of only elements.

    Additionally, only a small fraction of these compounds have their basic properties determined [ ]. This problem is even more complex for the infinitely large dataset formed by the all possible combinations of surfaces, interfaces, nanostructures, and organic materials, in which the complexity of materials properties is much higher.

    Therefore, it is reasonable to suppose that materials with promising properties are still to be discovered in almost every field [ ].

    In practice, several software packages and tools for different types of ML tasks exist, and are presented in table 3.

    General purpose codes work for the various types of problems section 2. Materials specific codes aid in the different steps of the MI workflow.

    These include data curation and representation by transforming general materials information compositional, structural, electronic, etc into feature vectors details in the next section 2.

    Table 3. Selection of materials informatics and machine learning codes and tools. Adapted from [ 22 ]. Finally, we now discuss an essential question regarding ML research: when ML should or not be employed and what kind of problems it tackles.

    An obvious crucial prerequisite is the availability of data, which should be consistent, sufficient, validated, and representative of the behavior of interest to be described.

    Once more we emphasize this requirement and thus, the common data generation process is generally better suited to traditional or HT approaches, at least initially.

    Additionally, one has to consider the strengths of machine learning methods, which can manage high-dimensional spaces in searching for relationships in data.

    The patterns discovered are then explicit encoded, rendering computational models that can be manipulated. In contrast, if human intuition can produce a physical model, ML is probably not needed by the problem.

    Therefore, ML methods are best suited to problems where traditional approaches have difficulties. Although it is not always clear to specify, if a problem can be identified into one of the general ML problem types described in section 2.

    Historically, areas which have questions with these characteristics have had successful applications of ML methods, such as in automation, image and language processing, social, chemical and biological sciences, and in recent times many more examples are emerging.

    Based on these characteristics, we glimpse on the common types of materials science applied problems which make use of data-driven strategies, and that are exemplified in section 3.

    A related strategy is to replace the description of a very complex or expensive property that is somewhat known, at least for a small class of materials by a simpler ML model, rendering its calculation less expensive.

    If properly validated, this model can then predict the complex property for unknown examples, expanding the data set. In the context of materials discovery and design, this strategy can be employed as a form of extending the data set before the screening, where the initial expensive data leads to more data through the ML model, which can then be screened for novel promising candidates.

    Other problems use feature selection techniques to discover approximate models and descriptors, which aid in the phenomenological understanding of the problem.

    Another type of problem and perhaps the most abundant is the clear advantageous problems in which expensive calculations can be replaced by a much more efficient model, such as replacing altogether DFT calculations for ML models such as in obtaining atomistic potentials for MD simulations, predicting the value of different properties gap, formation and total energies, conductivity, magnetization, etc.

    The representation of materials is a crucial component determining the machine learning performance. Only if the necessary variables are sufficiently represented then the learning algorithm will be able to describe the desired relationship.

    A representation objective is to transform materials characteristics such as composition, stoichiometry, structure, and properties into a quantitative numerical list, i.

    These variables used to represent materials characteristics are called features , descriptors , or even fingerprints. A general guideline can be expressed by a variant of Occam's razor which is a paraphrase famously attributed to Einstein, a representation should be 'as simple as possible, but not simpler'.

    For any new ML problem, the feature engineering process is responsible for most of the effort and time used in the project [ ].

    Other helpful characteristics for representations are having a high target similarity similarity between representation and original represented function , and from a computational perspective, functions of fixed dimensionality, and smooth and continuous, which ensures differentiability.

    These requisites presented act to assure that the models will be efficient with only the essential information. Such field has shown a relative degree of success, but also inconsistent performance of its models, arising from a lack of either proper domain of applicability, satisfactory descriptors, or machine learning validation [ ].

    Generally, a material can be described in several ways, of increasing complexity degree, depending on each problem needs.

    The simplest way is using only the chemical features such as atomic element types and stoichiometric information, which involves no structural characterization, therefore being more general but less specific to distinct polymorphs which can present different properties.

    This kind of rough description manages to describe general trends among very different types of materials. In order to increase the description capability of the ML models, higher complexity can be handled by introducing more relevant information available [ ].

    For descriptors based on elemental properties [ , , , ] this involves including and combining elements properties and statistics of these such as the mean, mean absolute deviation, range, minimum, maximum and mode.

    Stoichiometric attributes can include the number of elements, fractions, and norms. Even beyond, ionic character [ , ] and electronic structure attributes [ , , ], fingerprints [ ] and statistics can be included, to account for more intricate relationships.

    Including the structural information of the high dimensional space of atomic configurations [ ] is not a simple task. Common structural representations are not directly applicable to computational descriptions.

    Materials, especially solids, are commonly represented by their Bravais matrix and a basis, including the information of the translation vectors and the atom types and positions, respectively.

    For machine learning purposes, this representation is not suitable due to not being unique. In case of structural input, the requisites presented above indicate that the chemical species and atomic coordinates should suffice for an efficient representation.

    As such, the models should preserve the systems symmetries such as translational, rotational, and permutational. Ultimately, the representation objective is to ensure accuracy comparable to or superior than quantum mechanics calculations, for a wide range of systems, but with reduced computational cost.

    These so-called structural fingerprints are increasingly used to describe the potential energy surfaces PES of different systems, leading to force fields for classical atomistic simulations with QM accuracy, but with computational cost orders of magnitude lower and also linear scaling n behavior with the number of atoms.

    Most of these potentials benefit from chemical locality, i. Notable examples are the Gaussian Approximation Potentials GAPs [ , ], Behler—Parrinello high-dimensional neural network potentials [ , ], and Deep Potential molecular dynamics [ ].

    Related to the structural representation, a scoring parameter to identify dimensionality of materials was recently developed [ ].

    There are also methods for structural similarity measurements improving upon the commonly used root-mean-square distance RMSD [ ], such as fingerprint distances [ , ], functional representation of an atomic configuration FRAC [ ], distance matrix and eigen-subspace projection function EPF [ ], and the regularized entropy match REMatch [ ], used with SOAP.

    There is a vast collection of descriptor proposals in the literature that include more complex representations than the simple elemental properties discussed above, ranging from molecule-oriented fingerprints to descriptors for extended materials systems and tensorial properties.

    We now present a relatively chronological list used in recent materials research, which is considerable but not exhaustive.

    An important open discussion regards the interpretability [ , ] of the descriptors and consequently of the models obtained with ML [ , ].

    As already stated, one of the materials science objectives is to discover governing relationships for the different materials properties, which enable predictive capacity for a wide materials space.

    A choice can be made when choosing and designing the descriptors to be used. When prediction accuracy is the main goal, ML methods can be used as a black-box, and the descriptor interpretation and dimensionality are secondary.

    On the other hand, if the goal in addition to accuracy is understanding, physically meaningful descriptors can provide insight into the relationship described, and help even to formulate approximate and rough phenomenological models [ ].

    This cycle is presented in figure The revised connectivity between the four science paradigms. From empirical data to fundamental theories, which are realized in computational simulations, generating even more data.

    Statistical learning in turn can obtain simple phenomenological models that aids in theoretical understanding. In the ML case, the debate questions whether ML models can be purely interpolative closer to the 1st empirical science paradigm or also extrapolative closer to 2nd fundamental theoretical science paradigm , predicting more fundamental relationships beyond the given data class.

    Recently, Sahoo et al presented a novel approach capable of accurate extrapolation, by identifying and generalizing the fundamental relations to unknown regions of the parameter space [ ].

    No consensus exists about this discussion, and advances in research can make this debate obsolete. A pragmatic view on the causation versus correlation debate is to acknowledge that while discovering the underlying physical laws is the ideal goal, it is not guaranteed to happen.

    Otherwise, obtaining association patterns can be done much more quickly and could be an acceptable substitute for many practical problems [ 8 ].

    We discussed ways that machine learning can be used to directly predict materials properties or even for the discovery of novel materials.

    Another broader strategy is that ML methods can also be used to bypass or replace the calculations necessary to obtain the data in the first place.

    Here we briefly discuss the use of ML to extend and advance current methods for a variety of problems. Works in this direction have a broader interface with physics in general, developing methods applicable and inspired by different areas.

    There are several strategies that can be employed to circumvent the expensive Schrödinger equations calculations and optimize computational resources by using ML, without sacrificing accuracy.

    The general idea is presented in figure 16 left. A prominent example and intuitive approach is using ML to predict novel density functionals to be used within DFT, which can be readily used with current implementations [ , — ].

    The functionals to be predicted can be the exchange-correlation as used in the traditional DFT KS mapping, or of the orbital-free type.

    These three forms of mapping are presented in figure 16 right. Left An alternative to costly theoretical calculations by more efficient ML predictions.

    Reproduced from [ ]. For molecular dynamics simulations acceleration, ML was used to predict the properties of configurations already evaluated and similar ones, leaving only the expensive calculations of unseen configurations to be made on-the-fly [ , ].

    When referring to the ML training process, the datasets generation can be done with active learning [ ] instead of more traditional approaches like MD or metadynamics [ ].

    Quantum 'intuition' can also be incorporated in the ML training process by using a density-functional tight-binding DFTB or other model processing layer in neural networks [ ].

    Wider ML applications include obtaining corrections for non-covalent interactions [ , ], finding transition states configurations [ ] in a more efficient way than the nudged elastic band NEB method, as well for determining parameters for semiempirical quantum chemical calculations [ ] and DFTB [ ] models.

    Machine learning has also been used to obtain tight-binding like approximations of Hamiltonians [ ], solving the quantum many-body problem [ — ] and Schrödinger equation [ ] directly.

    The applications in physics also involve the important problems of partition functions determination [ ], finding phase transitions and order parameters [ — ], and obtaining the Green's function of models [ ].

    These examples show promising strategies to extend the frontiers of materials science research, which can be applied to study a variety of systems and phenomena.

    In the previous section 2 , we provided the basics of the approaches, presenting why and how they are used.

    In the following sections, we present a selection of works and several references that effectively illustrate how these approaches can be used for a variety of problems in materials science.

    We used DFT as a representant of the general class of methods used to generate data, due to being the most used method to materials science.

    The data, irrespective of where it came from, is then used in the HT and ML approaches. Much has been written on DFT applications, and articles and reviews of general [ ] and specific scope are constantly seen.

    DFT has been used for almost every kind of system ranging from atomic [ ], molecular [ , ], and chemical systems, extended solids, surfaces [ ], defects [ ], 0D [ — ], 1D [ — ], and 2D [ ] systems.

    The HT methods for novel materials discovery are directly related to the generation and storage of massive amounts of data.

    This data availability most theoretical databases are open access to the general scientific community is an important collaborative strategy for accelerating the discovery of innovative applications.

    In table 2 some examples of the largest databases are highlighted. These theoretical and experimental databases have been used for several applications: battery technologies, high entropy alloys, water splitting, high-performance optoelectronic materials [ ], topological materials, and others.

    Here we show some examples of its usage. We choose to focus mainly on the usage of large databases. Nevertheless, several groups generate their own databases, not relying only on those reported at table 2.

    Castelli et al screened materials from the Materials Project for solar light photoelectrochemical water splitting materials.

    Materials Project is fully calculated with PBE, i. With the improved bandgap description they created a descriptor based on the materials stability, band gap in the visible light region, and band edges alignment.

    The simplest definition of high entropy alloys HEAs is based on the number and concentration of its components and on the formation of a single phase solid solution.

    Some authors have more restrict definitions based also in its micro-structural arrangement [ ]. These HEAs have attracted attraction recently, due to its promise of high stability against precipitations of its components.

    Precipitation is undesirable because it may modify the properties of the alloys. The mechanism behind HEAs stability relies on its high entropy, which will result in a dominance of the entropic TS term over the enthalpic H one in the Gibbs free energy.

    With an effect of avoiding phase separation, a solid solution will be formed. The existent models to predict phase transition in HEAs are, in general, unsatisfactory.

    The main reason is the absence of experimental and theoretical data. The combinatorial rules are onerous and a 5 component HEA with an 8 atom unit cell would require more than , DFT total energy calculations.

    The LTVC model is a combination of HT DFT calculations, performed in a small configurational subspace, followed by cluster expansion calculations to increase the energetics data availability and mean field statistical analysis.

    Finally, an order parameter is proposed to determine possible phase transitions. Thermoelectric materials are able to generate electrical current via a temperature gradient.

    A promising application is to recover dissipated energy heat. The last variable, in general, has an electronic and lattice contribution.

    DFT is able to calculate the components of the ZT. Nevertheless, its computation is extremely costly, since it requires a fine sampling of the reciprocal space [ , ].

    Only recently HT investigations of thermoelectric materials were feasible, owing to interpolation schemes capable of circumventing the computational cost [ 74 — 77 ].

    They also found a direct relation between the power factor and the materials band gap. Bhattacharya et al explored alloys as possible novel thermoelectric materials [ ].

    Chen et al performed calculations over entries on the Materials Project database [ ]. They found a good agreement between experimental and theoretical Seebeck coefficients.

    Nevertheless, the power factor is less accurate. They also determined correlations between the crystal structure and specific band structure characteristics valley degeneracy that could guide materials modifications for enhanced performance.

    Identification of suitable optoelectronic materials [ ] as well as solar absorbers [ , ] have also been possible via HT calculations.

    Another important study based on HT methods is the obtention of elastic properties of inorganic materials [ 18 , ] and the subsequent structuring of the data into publicly available databases.

    Additionally, Mera Acosta et al [ ] performed a screening in the AFLOWLIB database, showing that three-dimensional materials can exhibit non-magnetic spin splittings similar to the splitting found in the Zeeman effect.

    Topological materials can be classified into topological insulators TIs , topological crystalline insulators, topological Dirac semimetals, topological Weyl semimetals, topological nodal-line semimetals, and others [ , — ].

    The topologically nontrivial nature is tied to the appearance of inverted bands in the electronic structure. For most topological materials, band inversions have been demonstrated to be induced by delicate synergistic effects of different physical factors, including chemical bonding, crystal field and, most notably, spin—orbit coupling SOC [ , , ].

    Indeed, the search for new TIs was thus guided by experience and intuition. For example, in two-dimensional materials the search was initially focused on heavy elements with high SOC [ , — ].

    The search for novel topological insulators is an example of the fundamental role of the computational simulations in the prediction of new materials and devices design.

    DFT has been essential to understand physical and chemical phenomena in TI materials. One of the usual approaches to predict TIs starts by selecting materials isoelectronic to the already known TIs and then, employing DFT calculations, verify if the proposed materials feature band inversions at symmetry protected k -points or non-zero topological invariants.

    These calculations typically have a high computational cost, and hence, this trial-and-error process is not usually feasible.

    In the seminal work of Yang et al , it was shown that semi-empirical descriptors can aid the selection of materials, allowing the efficient use of HT topological invariant calculations to predict TIs [ ].

    The proposed descriptor represents the derivative of the bandgap without SOC with respect to the lattice constant [ ], requiring the band structure calculation at various values of the lattice constant.

    Thus, material screening and high-throughput calculations were combined to study the bandgap evolution as a function of the hydrostatic strain.

    These semi-empirical descriptors capture the evolution of the states involved in the band inversion for a given compound. The authors thus predicted 29 novel TIs.

    In order to avoid the complex calculations of the topological invariants, a simple and efficient criterion that allows ready screening of potential topological insulators was proposed by Cao et al [ ].

    A band inversion is typically observed in compounds in which the SOC of the constituent elements is comparable with the bandgap.

    The validity and predictive power of such criterion were demonstrated by rationalizing many known topological insulators and potential candidates in the tetradymite and half-Heusler families [ , ].

    This is an unusual example since the use of atomic properties for the prediction of complex properties has only been extensively explored through ML techniques, such as the SISSO method See section 3.

    Despite the great influence that has had the understanding of the nontrivial topological phases in condensed matter physics and the great efforts to find novel TI candidates, the predicted systems are reduced to few groups of TIs.

    For instance, only 17 potential TIs were identified by carrying out HT electronic band structure calculations for 60, materials [ ].

    Using novel approaches, this problem was recently addressed by three different works [ — ], in which thousands of compounds have been predicted to behave like TIs.

    Here we will briefly discuss these works. In the first work, Bradlyn et al [ ] put forward the understanding of topologically protected materials by solving a general question: 'Out of , stoichiometric compounds in material databases, only several hundred of them are topologically nontrivial.

    Are TIs that esoteric, or does this reflect a fundamental problem with the current piecemeal approach to finding them? This theory gives a complete description of periodic materials, unifying the chemical orbitals described by local degrees of freedom and band theory in the momentum space [ , ].

    The recently proposed elementary band representations are an example of a general descriptor to perform materials screening, however, details related to the atomic composition require the band structure calculation.

    A feature space including the elementary band representations could be a strategy to find ML-based models for novel hypothetical TI candidates.

    The authors designed what is known as the first catalog of topological electronic materials. The authors also found topological semimetals with the band crossing points located near the Fermi level [ ].

    Finally, Choudhary et al performed HT calculations for the SOC spillage, a method for comparing wave functions at a given k -point with and without SOC, reporting more than high-spillage TIs candidates [ ].

    The authors extended the original definition of the spillage, which was only defined for insulators [ ], by including the number of occupied electrons n occ k , i.

    Thus, this screening method is not only suitable to identify topological semimetals, but is also applicable to the investigation of disordered or distorted materials.

    We consider that the prediction of new TIs has been one of the greatest contributions and victories of HT methods and materials screening.

    In spite of these great advances, there is still a very long route for the total comprehension of phenomena in non-trivial topological states and the discovery of materials presenting phases not yet investigated.

    The 2D materials era was initiated with the graphene isolation by Novoselov and Geim [ ]. Graphene has shown how quantum confinement can significantly alter the 2D allotrope in comparison with its 3D counterpart.

    Posterior to discovery of graphene a profusion of 2D materials have been proposed and synthesized: transition metal dichalcogenides TMDC , h -BN, silicene, germanene, stanene, borophene, II-VI semiconductors, metal oxides, MXenes, and many others, including recently non van der Waals materials [ — ].

    The first approach using data-mining and HT calculations to discover novel 2D materials was performed by Björkman et al [ , ]. Using a descriptor based on symmetry, packing ratio, structural gaps and covalent radii they screened the ICSD database [ ] and 92 possible two-dimensional compounds were identified.

    The interlayer binding energy, which is closely related to the exfoliation energy, was calculated using a very accurate scheme based on nonlocal correlation functional method NLCF [ ], the adiabatic-connection fluctuation-dissipation theorem within the RPA [ ] and different van der Waals functionals [ — ] along with the traditional LDA and GGA functionals.

    Despite their pioneer work, the results were still communicated in the traditional narrative form.

    Only recently, the construction of large databases of 2D materials became popular. In general, these databases are constructed via DFT calculations using as prototypes experimental information.

    In the next few lines, we will briefly describe some of these 2D databases and their construction strategies. Choudhary et al made publicly available a 2D database with hundreds of single-layered materials [ ].

    The PBE functional is known to overestimate the lattice constant. This overestimation is larger for van der Waals systems. For example, PBE is unable to describe the graphite structure, since there is no energy minimum as a function of the interlayer distance [ ].

    After this initial screening, they computed the exfoliation energy with proper vdW functionals, to identify possible 2D candidates.

    This simple descriptor correctly predicts layered materials Another distinct feature of this database is the large plane wave cut-off and reciprocal space sampling.

    The TSA first calculates the materials bonding, based on covalent radii, to identify atoms clusters. If only one cluster is present the structure is unlikely to be layered.

    If TSA finds clusters structures the supercell is increased n times in each direction and a new search for clustering is performed. If the cluster number of atoms increases quadratically with n , the system is layered.

    Mounet et al [ 3 ] used an algorithm in the same spirit of the TSA approach to search for layered compounds. Further, they calculated vibrational, electronic, magnetic and topological properties for a subset of materials.

    They found 56 magnetically ordered systems and two topological insulators. Haastrup released one of the largest 2D databases [ ] with more than materials.

    The adopted strategy is different from the previous databases. They implemented a combinatorial decoration approach of known crystal structure prototypes, for more than 30 different ones.

    The thermodynamic stability is determined via the convex hull approach. They used information about the formation energy and phonon frequencies of known 2D materials to conceive a stability criterion.

    The prototypes are classified as having a low, medium and high stability depending on its hull energy and dynamical matrix minimum eigenvalue.

    Other calculated properties include elastic, electronic, magnetic including magneto-crystalline anisotropy , and optical properties.

    They also employed, in a smaller subset, more sophisticated schemes such as hybrid functionals, GW approximation, and RPA calculations.

    These databases are now being screened for different properties. Ashton et al [ ] discovered a new family of Fe-based large spin-gap as large as 6. Four new topological insulators have been predicted by Li et al [ ] screening 2D materials of the Materials Web database.

    The largest gap found was 48 meV for TiNiI. Olsen et al discovered several 2D nontrivial materials including topological insulators, topological crystalline insulators, quantum anomalous Hall insulators and dual topological insulators which possess time reversal and mirror symmetry [ ].

    The 3D databases are well established and have been used widely. In contrast, the number of works using the proposed 2D databases is relatively small.

    This current status provides great opportunities for further exploration in the near future. In this section we present a selection of research applying machine learning techniques to materials science problems, illustrating the materials informatics capabilities explored in the literature.

    The research questions studied involve different types of ML problems as described generally in section 2. As the application of ML techniques to materials problems is relatively recent, articles, perspectives, and reviews are nowadays increasingly emerging in the literature.

    Some works that illustrate the ML concepts and examples applied to diversified materials problems are given by [ 22 , , , — ].

    We therefore focus here on selected examples which present recent advances to this area. A common topic for ML applied to materials research is the accelerated discovery of compounds guided by data.

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