3D Equivariant GNNs

Module 3 expands our scope from 2D images to 3D Point Clouds and Graphs. This is where methods become crucial for computational sciences like chemistry, biology, and physics.

1. Motivation: Science vs. Cats

In standard computer vision (e.g., classifying cats), we deal with 2D images on regular grids. However, in scientific domains, data rarely comes on a grid:

• Chemistry: Molecules are collections of atoms with 3D positions and chemical bonds. They are naturally modeled as Graphs.
• Physics: N-body simulations involve particles interacting in continuous 3D space.
• Medical Imaging: Diffusion MRI data represents fiber orientations at each voxel, which can be seen as a signal on the sphere $S^2$.

2. The Need for Equivariance

While Graph Neural Networks (GNNs) can handle the irregular structure of this data, standard GNNs often ignore the geometric nature of the features. Equivariance is non-negotiable here:

• Invariance: The potential energy of a molecule must be the same regardless of how we rotate the coordinate system.
• Equivariance: If we predict forces acting on atoms, rotating the molecule should result in the force vectors rotating exactly the same way.

3. The Toolkit

To build these networks, we move from the 2D rotation group $SO(2)$ to the 3D rotation group $SO(3)$. The mathematical tools become more complex but more powerful:

• Group: $SO(3)$ (3D Rotations) and $SE(3)$ (3D Roto-translations).
• Representations: Wigner-D matrices $D^l(g)$.
• Interaction: The Clebsch-Gordan Tensor Product.

This module will guide you through building these powerful 3D equivariant architectures, often referred to as Tensor Field Networks or Steerable E3-GNNs.