Materials informatics seeks to establish structure—property relationships in a high-throughput, statistically robust, and physically meaningful manner Lookman et al. Researchers are seeking connections in materials datasets to find new compounds Hautier et al. To these ends, scientists are increasingly utilizing machine learning, which involves the study and construction of algorithms that can learn from and make predictions on data without explicit human construction.
Those algorithms can be as simple as an ordinary least squares fit to a data set or as complicated as the neural networks used by Google and Facebook to connect our social circles. In materials science, for example, researchers have used LASSO least absolute shrinkage and selection operator to construct power series e.
Tree-based models are being used to optimize 3D printed part density Kamath, , predict faults in steel plates Halawani, , and select dopants for ceria water splitting Botu et al. Clustering along with principal component analysis PCA has been used to successfully reduce complex, multidimensional microscopy data to informative local structural descriptions Belianinov et al. There are many more possibilities for using machine learning methods in materials informatics, but they possess risks in misapplication and interpretation if adapted from other data problems without precautions.
In this paper, we describe nuances associated with using machine learning and how theoretical domain-based understanding serves as a complement to data techniques. We start with the problem of overfitting to data and some ways seemingly minor choices can change our understanding and confidence in predictions. Then, we discuss rational choices for materials descriptors and ways to produce them. Lastly, we examine the importance of producing simple models and provide a sample workflow for the theoretician. In machine learning, overfitting occurs when a statistical model accurately fits the data at hand but fails to describe the underlying pattern.
This can lead to inaccurate predictions for novel compounds or structures and will also often make physical interpretability difficult due to an excess of model parameters. A way to combat overfitting is to keep separate datasets for training a model and for testing it.
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In materials science, however, data can be expensive and laborious to obtain and keeping a set amount off-limits is anathema. Data scientists, therefore, usually divide the data into equal partitions, using a fraction of the data to test model performance on and use the rest for training in a process called k -fold cross validation Stone, The partitions are then iterated over so every partition has been used as the test set once and the errors are averaged. This has the effect of simulating how accurately a model will handle new observations. Note, sometimes zero mean and unit variance are required for feeding features into scale-sensitive algorithms such as linear regression or PCA.
The calculation of means and standard deviations should be reserved for after partitioning so as to avoid information contamination. Overfitting is generally more a problem for datasets with few samples relative to features. Unfortunately, this under constrained problem often applies in theory-guided models where relatively few materials have been thoroughly characterized either experimentally or computationally. Care must be taken in choice of basis set, model parameters, and model selection parameters [hyperparameters Cawley and Talbot, ] to qualitatively find the correct model.
Unfortunately, even with cross-validation, model error estimates can sometimes be overly optimistic. For severely under constrained problems, Bayesian error estimation methods may be called for Dalton and Dougherty, Another subtle but important detail concerns the underrepresentation of certain classes in data when performing classification.
For example, materials scientists are often interested in identifying uncommon properties, like high T C superconductivity or large Z T for improved thermoelectric power. Luckily, machine learning practitioners have dealt with these issues for some time, and there are ways to mitigate the problem. Techniques mainly focus on resampling methods for balancing the dataset and adding penalties to existing learning algorithms for misclassifying the minority class. Of course, one remedy data scientists often ignore is to collect more data, which can be achieved in practice by a materials scientist.
Decision trees are a machine learning technique known to be prone to these sorts of problems, and we use them next as an example to explore the nuances in more detail. Decision trees operate by recursively partitioning the data with a series of rules designed according to an attribute value test Quinlan, The end result is analogous to a flow chart with levels of rule nodes leading to different predictions. Nodes appearing earlier on in the tree separate more samples than lower ones and can be viewed as more important in the stratification procedure. What may be less well appreciated by new users using machine learning approaches is that by simply changing the criterion used for selecting the partitions, one can observe qualitatively different results among the decision trees.
For example, we used data from a recent study on predicting high-temperature piezoelectric perovskites Balachandran et al. Figure 1.
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Decision trees for determining perovskite formability based on gini impurity top and Shannon entropy or information gain bottom using data from Balachandran et al. At each node, a check occurs and if true proceeds to the left and vice-versa. Domain knowledge tells us that the tolerance factor, difference in ionic radii ratios, and A-site ionicity are proportional to the radius of the A -site cation, and are therefore expected to originate from close packing preferences of ionic solids.
The GII is dependent on the difference between ideal and calculated bond valences, which can capture local bonding effects in addition to the steric packing preferences. Importantly, if the GII has useful predictive power for the cubic perovskite oxide stability relative to the binary oxide phases, which it may decompose into, more compounds could be screened than if only the radius based factors were predictive. Indeed, there are many cations with known bond valence parameters but lacking the necessary tabulated coordinate ionic radii that may be used in the tolerance factor calculation.
In addition, the incorporation of the GII in the model could elucidate additional bonding characteristics that lead to phase stability.
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These trees were trained on the same data, so one would assume the same underlying physics should be captured by both, but that is not entirely the case. Building an optimally accurate tree is computationally expensive, and so heuristic algorithms are used instead, which are not guaranteed to find a global solution. Indeed, the resulting tree can vary even between multiple runs of the same algorithm. The high level of accuracy in both cases indicates that a handful of structural features, such as ionic radii ratios and ideal A—O bond distance, are suitable to assess if an AB O 3 composition will the form the perovskite structure.
Cross-validation is an optimistic guess that only works if the data supplied appropriately samples the underlying population.
We are not saying here that one tree is definitely wrong and one is definitely right, but rather that any given model found in the literature is the result of numerous choices on behalf of the modeler. Materials informatics trades in physically meaningful parameters. So-called descriptors of materials properties are key to making predictions and building understanding of systems of interest Rondinelli et al.
Some properties of a good descriptor have been laid out previously Ghiringhelli et al. Namely, a good descriptor should be simpler to determine than the property itself, whether it is computationally obtained or experimentally measured. It should also be as low-dimensional as possible, and uniquely characterize a material. Descriptors in materials science can come from a variety of levels of complexity.
Atomic numbers, elemental groups or periods, electronegativities, and atomic radii can be read off periodic tables and used to predict structure type. Densities and structural parameters can be measured in an experiment for the purposes of predicting mechanical properties. And of course, combinations of quantities from the same or different levels can be descriptors as well. There is, however, no universally acknowledged method for choosing descriptors. Descriptor choice will depend heavily upon the phenomena being studied.
For instance, atomic radii happen to be important in predicting bulk metallic glass formation Inoue and Takeuchi, as well as perovskite formability Balachandran et al. However, attempting to use covalent radii for both will miss the important ionic character of the atoms in perovskite systems. Brgoch et al. An insight came from reading the literature on molecular phosphors, where structural rigidness played a key role in photoemission yield.
With this knowledge, the authors were able to construct a descriptor for photoluminescent quantum yield in solids based on the Debye temperature, related to the stiffness of the vibrational modes, and band gap. A similar recognition of the underlying physics yielded a descriptor for carrier mobility in thermoelectrics based on bulk modulus and band effective mass Yan et al. One review Curtarolo et al. However, it remains a challenge to motivate further exploration without an underlying theoretical justification.
In some cases, the best model may not be capable of being built from the features initially selected. A simple example might be predicting an activation energy from observed diffusion measurements using regression analysis. Using the natural log of diffusion constants yields a better fit than fitting the raw values. Depending upon the material system of interest one can enumerate as many physically plausible primary descriptors as possible and then generate new descriptors from them in some manner.
This could include groupings from dimensional analysis Rajan et al.
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In all cases, care must be taken to avoid incompatible operations e. In image data, edges are often extracted as features from primary pixel data to be used in learning Umbaugh, Once features have been extracted, there might then be some downselection to test only the most important features. Ghiringhelli et al. Materials scientists are interested in establishing clear causal relations between materials structure defined broadly across length scales and properties.
While a model employed by Netflix might be evaluated solely in terms of predictive accuracy and speed, scientific models have further constraints such as a minimal number of parameters and adherence to physical laws. If a model cannot be communicated clearly except from computer to computer, its contribution will be minimal. It is the obligation of the modeler to translate the results of their work into knowledge other materials scientists can use in aiding materials discovery or deployment. That being said, eliminating parameters by hand to make an intelligible model is often impractical.
In this case, there are some helpful techniques available. Principal component analysis is a powerful technique for data dimensionality reduction. In essence, PCA is a change of basis for your data with the new axes [principal components PCs ] being linear combinations of original variables. Each principal component is chosen so that it lies along the direction of largest variance while being uncorrelated to other PCs. When the data are standardized to have zero mean, these PCs are eigenvectors to the covariance matrix of the samples.
Although perfect description of the original data requires as many PCs as original features in theory, typically, some number of PCs is selected for retention based upon a threshold amount of variance explained or correlation with a feature of interest. However, care must be taken to judiciously apply PCA. Authors will readily acknowledge PCs are not necessarily simple to interpret physically especially with image data Belianinov et al.
Figure 2. The ideal subspace produced with PCA is shown in black. Next, the net atomic displacements involved in each mode, which are required to reach the observed structure, are obtained as a root-sum-squared displacement magnitude in angstroms through this structural decomposition. Digital subscriptions are available across all devices and include all regular issues released during your subscription.
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