  Machine learning (ML) methods are becoming integral to scientific inquiry in
numerous disciplines, such as material sciences. In this manuscript, we
demonstrate how ML can be used to predict several properties in solid-state
chemistry, in particular the heat of formation of a given complex
crystallographic phase (here the $\sigma-$phase, $tP30$, $D8_{b}$). Based on an
independent and unprecedented large first principles dataset containing about
10,000 $\sigma-$compounds with $n=14$ different elements, we used a supervised
learning approach, to predict all the $\sim$500,000 possible configurations
within a mean absolute error of 23 meV/at ($\sim$2 kJ.mol$^{-1}$) on the heat
of formation and $\sim$0.06 Ang. on the tetragonal cell parameters. We showed
that neural network regression algorithms provide a significant improvement in
accuracy of the predicted output compared to traditional regression techniques.
Adding descriptors having physical nature (atomic radius, number of valence
electrons) improves the learning precision. Based on our analysis, the training
database composed of the only binary-compositions plays a major role in
predicting the higher degree system configurations. Our result opens a broad
avenue to efficient high-throughput investigations of the combinatorial binary
calculation for multicomponent prediction of a complex phase.
