Fig. 3: The overview of the sub-networks in HVNet and HTNet.

a is the architecture of the sub-network, termed relational message passing convolutional (RMConv) layer. Such a RMConv layer is a simplified and modified PAINN²² invoked for a specific type of interaction, and is constituted by (b) radial message layer, c radial update layer, d angular message layer, and e or f angular update layer. {s} is the set of scalar node features and initially set as the atomic numbers, which will be passed through an embedding layer. Initial vectorial node feature \({\overrightarrow{v}}^{(0)}\) is an all-zero vector with a fixed dimension. \(\sin \left(\frac{n\pi }{{r}_{{{\rm{cut}}}}}\parallel {\overrightarrow{r}}_{ij}\parallel \right)/\parallel {\overrightarrow{r}}_{ij}\parallel\) with 1 ≤ n ≤ 30 are selected as radial basis functions (RBF)²⁰ and a cosine cutoff f_cut⁵⁹ is also adopted in the filter. The original message layer in PAINN²² (i.e., a MPNN layer) is decomposed into (b) and (c) (i.e., radial message layer and radial update layer). A modified and simplified update layer is decomposed into (d) and (e) for HVNet or (f) for HTNet. These layers model three-body interactions via expressing angular information explicitly.