Fig. 6: The architecture of PaEGNN.
a Illustration of the multi-graph representation designed to capture atomic interactions across cell boundaries in periodic structures. b Message passing that collects messages from a node’s neighbors and message updating that updates node representations using a node’s internal states. c Overview of PaEGNN, comprising four layers, each with message passing and message updating phases, taking unit cell offset integer Kji, initial vector \({\overrightarrow{{{\boldsymbol{x}}}}}_{i}^{(0)}=\overrightarrow{{{\mathbf{0}}}}\), initial scalar \({{{\boldsymbol{x}}}}_{i}^{(0)}=E({z}_{i})\), and relative position \({\overrightarrow{{{\boldsymbol{r}}}}}_{ji}\) as inputs and outputting the final vector \({\overrightarrow{{{\boldsymbol{x}}}}}_{i}^{(T)}\) and scalar \({{{\boldsymbol{x}}}}_{i}^{(T)}\) representations. d During the message passing phase, a node vi aggregates messages from neighboring vectors \({\overrightarrow{{{\boldsymbol{x}}}}}_{j}^{(t)}\) and scalars \({{{\boldsymbol{x}}}}_{j}^{(t)}\), forming intermediate vector and scalar variables \({\overrightarrow{{{\boldsymbol{m}}}}}_{i}\) and mi. e The message updating phase integrates the F vectors and F scalars within \({\overrightarrow{{{\boldsymbol{m}}}}}_{i}\) and mi to generate updated vector \({\overrightarrow{{{\boldsymbol{x}}}}}_{i}^{(t+1)}\) and scalar \({{{\boldsymbol{x}}}}_{i}^{(t+1)}\).
