  Message-passing neural networks (MPNN) have shown extremely high efficiency
and accuracy in predicting the physical properties of molecules and crystals,
and are expected to become the next-generation material simulation tool after
the density functional theory (DFT). However, there is currently a lack of a
general MPNN framework for directly predicting the tensor properties of the
crystals. In this work, a general framework for the prediction of tensor
properties was proposed: the tensor property of a crystal can be decomposed
into the average of the tensor contributions of all the atoms in the crystal,
and the tensor contribution of each atom can be expanded as the sum of the
tensor projections in the directions of the edges connecting the atoms. On this
basis, the edge-based expansions of force vectors, Born effective charges
(BECs), dielectric (DL) and piezoelectric (PZ) tensors were proposed. These
expansions are rotationally equivariant, while the coefficients in these tensor
expansions are rotationally invariant scalars which are similar to physical
quantities such as formation energy and band gap. The advantage of this tensor
prediction framework is that it does not require the network itself to be
equivariant. Therefore, in this work, we directly designed the edge-based
tensor prediction graph neural network (ETGNN) model on the basis of the
invariant graph neural network to predict tensors. The validity and high
precision of this tensor prediction framework were shown by the tests of ETGNN
on the extended systems, random perturbed structures and JARVIS-DFT datasets.
This tensor prediction framework is general for nearly all the GNNs and can
achieve higher accuracy with more advanced GNNs in the future.
