# Elasticity predictions

# Introduction

The Materials Project (MP) provides a growing collection of elastic constants calculated from first principles Density Functional Theory (DFT). Please see Elasticity calculations for details regarding the MP elasticity workflow [1].

For compounds that have not yet been processed with the elasticity workflow, the MP offers statistical learning (SL) predictions of the Voigt-Reuss-Hill [2] average bulk and shear moduli (K_{VRH} and G_{VRH}, respectively). The SL models were trained with a diverse set of 1,940 k-nary compounds, using the K_{VRH} and G_{VRH} moduli calculated by the elasticity workflow.

# Formalism

Ensemble statistical learning techniques construct a predictor from a collection or ensemble of weak learners. Each weak learner is either a single descriptor or a function of just a few descriptors, which limits the level of interaction between descriptors. Gradient boosting (GB) is a very flexible ensemble technique, which makes few assumptions regarding the form of the solution and iteratively builds a predictor from a series of weak learners while minimizing the residual of a loss function [3]. GB implementations use regularization techniques to reduce the risk of over-fitting, which typically include limiting the level of interaction between descriptors, limiting the number of iterations per some risk criteria, and employing shrinkage [3]. At each iteration, the weak learner that causes the greatest reduction in the loss function’s residual is selected and added to the model, however, when shrinkage is employed, each new term is attenuated by the learning rate.

# Descriptors

The successful application of SL requires a set of descriptor candidates that sufficiently explain the diversity of the phenomenon being learned. We distinguish between composition and structural descriptors. Composition descriptors are calculated from elemental properties and only require knowledge of a compound’s composition. Structural descriptors require knowledge of a compound’s specific structure and are calculated using DFT.

# Citations

To cite elastic constant predictions within the Materials Project, please reference the following works:

- "de Jong M, Chen W, Notestine R, Persson K, Ceder G, Jain A, Asta M, and Gamst A (2016) A Statistical Learning Framework for Materials Science: Application to Elastic Moduli of k-nary Inorganic Polycrystalline Compounds, Scientific Reports 6: 34256." doi:10.1038/srep34256
- "de Jong M, Chen W, Angsten T, Jain A, Notestine R, Gamst A, Sluiter M, Ande CK, van der Zwaag S, Plata JJ, Toher C, Curtarolo S, Ceder G, Persson KA, Asta M (2015) Charting the complete elastic properties of inorganic crystalline compounds. Scientific Data 2: 150009." doi:10.1038/sdata.2015.9

# References

[1] de Jong M, Chen W, Angsten T, Jain A, Notestine R, Gamst A, Sluiter M, Ande CK, van der Zwaag S, Plata JJ, Toher C, Curtarolo S, Ceder G, Persson KA, Asta M (2015) Charting the complete elastic properties of inorganic crystalline compounds. Scientific Data 2: 150009.

[2] Hill, R. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society. Section A 65, 349 (1952).

[3] Hastie, T., Tibshirani, R. & Friedman, J. The elements of statistical learning: data mining, inference, and prediction. (Springer, 2011), second edn.

# Authors

- Randy Notestine
- Maarten de Jong