Regular GNNs

Alternatively to Steerable methods, we can use Regular GNNs. These methods rely on scalar features and invariant attributes (like distances and angles) to achieve equivariance.

1. Invariant Message Passing (e.g., SchNet)

The simplest approach is to condition messages only on pairwise distances $\|x_j - x_i\|$. Since distance is invariant to rotations, the whole network becomes rotation invariant. However, this loses directional information (all neighbors at distance $r$ are treated the same).

2. Directional Message Passing (e.g., DimeNet, GemNet)

To recover directional information without using steerable vectors, we can use higher-order invariants like angles (triplets of atoms) and torsion angles (quadruplets).

• DimeNet: Performs message passing between directed edges rather than nodes. This allows defining an angle between the incoming edge $kj$ and outgoing edge $ji$.
• GemNet: Extends this to include torsion angles and two-hop neighbors, effectively performing "regular group convolution" on the space of positions and orientations.

3. Lifting Approaches (e.g., LieConv)

Another strategy is to Lift the input to the full group $SE(3)$ (Position + Orientation) and perform a standard group convolution on this lifted space. This is mathematically equivalent to regular group convolution but implemented on point clouds.