Predicting elastic stiffness tensor components for 2D materials using machine learning
This competition is part of the course
[10316 Materials design with machine learning and artificial intelligence](https://kurser.dtu.dk/course/10316)
at DTU January 2022. During the first two weeks you have learned the basics of programming and machine learning, and now it's time to apply these skills in a real world problem.
The objective of the competition is to build a machine learning model that predicts the elastic tensor components of 2D materials.
For elastic materials, the linear elasticity can be described by the general equation for Hooke’s law:
\[ \boldsymbol{\sigma} = \boldsymbol{C} : \boldsymbol{\varepsilon} \]
or in engineering notation:
\[ \sigma_{ij} = C_{ijkl} \varepsilon_{kl}\]
where  \(  \boldsymbol{\sigma}  \) is the stress tensor, \(  \boldsymbol{\varepsilon}  \) is the strain tensor and \(  \boldsymbol{C}  \) is the fourth-order stiffness tensor. For 2D materials there are 3 unique relations between stress and strain:
\[\begin{align} \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} &= \begin{bmatrix} C_{1111} & C_{1122} & C_{1112} \\ C_{2211} & C_{2222} & C_{2212} \\ C_{1211} & C_{1222} & C_{1212} \end{bmatrix} \cdot \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix} \\ &= \begin{bmatrix} C_{1111} & C_{1122} & C_{1112} \\  & C_{2222} & C_{2212} \\ sym & & C_{1212} \end{bmatrix} \cdot \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix} \\ &= \begin{bmatrix} C_{11} & C_{12} & C_{13} \\  & C_{22} & C_{23} \\ sym & & C_{33} \end{bmatrix} \cdot \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix} \end{align}\]
The bulk modulus (Voigt definition) can be calculated as
\[ K_V = \frac{1}{4} (C_{11}+C_{22}+2C_{12})\]
which is the prediction objective for this competition.
The dataset for the competition consists of 3858 2D materials along with their DFT-calculated elastic components. The structures are provided as
[Atoms](https://wiki.fysik.dtu.dk/ase/ase/atoms.html)
objects containing the atomic numbers and positions for the structure. The dataset is divided into two sets (85% for training and 15% for testing). For the test set, only the structures and energy are provided (and not the elastic components).