Prediction of local electric polarization using graph neural networks

The calculation of electric polarization based on the Berry phase (BP) method plays a crucial role in understanding ferroelectric materials. However, it cannot capture local polarization within the supercell, especially in systems with complex topological structures such as flexoelectric structures and polar skyrmions. In this study, we present a method for calculating flexoelectric coefficients via graph neural networks (GNNs). We represent the polarization contribution of each unit cell to the overall supercell as the sum of the tensor contributions from each atom, while the tensor contribution from each atom can be represented as a linear combination of the local spatial components projected along the edge directions of a cluster centered on that atom. The local polarization can be obtained using GNNs without density-functional theory calculations. Using this method, we accurately calculate the flexoelectric coefficients in $\mathrm{BaTi}{\mathrm{O}}_{3}$ and Si, as well as the polarization distribution in the $\mathrm{PbTi}{\mathrm{O}}_{3}/\mathrm{SrTi}{\mathrm{O}}_{3}$ superlattice. Our method provides a practical approach for local contributions of electric polarization calculations in large-scale systems.