  Accurate predictions of defect formation energies are crucial to
understanding defect properties in materials, such as favorable defect types
and concentration, and to investigating the interplay between defects and other
functional properties of materials. To overcome the computational expense of
density-functional theory calculations in large supercells, several graph
neural network (GNN) models were proposed to predict the defect formation
energies. However, their performance is limited due to the fact that defect
formation energies depend strongly on the local atomic configurations near the
defect sites and to the over-smoothing problem of GNN. Herein, we demonstrate
that persistent homology features, which characterize the topological structure
of the local chemical environment around each atomic site, encode the
structural information of defects (such as vacancies and substitutions),
including the size of the defects, the number of defects near each atom, and
the distance to those defects. Using the dataset comprising a wide spectrum of
O-based perovskites with vacancies of various elemental type as an example, we
construct and train the GNN models to predict the neutral vacancy formation
energies, and show that incorporating the persistent homology features, along
with proper choices of the graph pooling operations, significantly increases
the prediction accuracy and overcomes the non-convergence issue with respect to
the supercell size in previous GNN models. Furthermore, using defective BaTiO3
with multiple substitutions and multiple vacancies as examples, our GNN model
can also predict the defect-defect interaction energies accurately. These
results suggest that persistent homology features are highly effective in
predicting defect-related properties and can be integrated into the vast family
of advanced graph neural network models for future defect studies.
