Geometric Architecture

A geometric architecture builds symmetry assumptions directly into a model. It is used when inputs include coordinates, directions, graphs, or physical structures.

The core design target is:

$$F(g\cdot x)=\rho (g)F(x)$$

where $g$ is a transformation such as rotation or translation, and $\rho (g)$ specifies how the output should transform.

The design starts by matching input type, target type, and group:

$$(\text{input object},\text{target},G)\to (\text{features},\text{messages},\text{readout})$$

For scalar prediction under rigid motions:

$$F(RX+1t^{\top })=F(X)$$

For coordinate prediction:

$$F(RX+1t^{\top })=RF(X)+1t^{\top }$$

These are different architecture contracts.

Design Choices

• Use invariant features such as distances for scalar outputs.
• Use equivariant coordinate or vector updates for structure prediction.
• Keep scalar, vector, and higher-order channels separate when needed.
• Use graph construction that respects the intended geometry.
• Match the final readout to the target: scalar score, vector field, coordinates, or distribution.

Architecture Patterns

• Distance-based GNN: invariant edge features, good for scalar readouts.
• Equivariant coordinate GNN: scalar messages plus vector coordinate updates.
• Tensor/irrep network: higher-order features that transform under SO(3) representations.
• Local-frame model: expresses geometry in learned or constructed local frames.
• Hybrid model: invariant encoder plus equivariant refinement or generative head.

Data Contract

• Coordinates should state source: experimental, predicted, docked, generated, simulated, or conformer-generated.
• Molecular inputs should state stereochemistry, protonation, tautomer, and conformer choices when relevant.
• Protein structures should state chain selection, missing residues, alternate locations, and residue alignment when relevant.
• Splits should match the generalization claim: ligand scaffold, protein family, complex pair, assay/source, or time.

Failure Modes

• The architecture is symmetric but graph construction uses leakage-prone context.
• The target is scalar but preprocessing centers on a known ligand pose unavailable at deployment.
• Reflection symmetry is enforced on chirality-sensitive chemistry.
• Batch-level pooling mixes multiple molecules or complexes.

Checks

• Is the target invariant or equivariant?
• Should reflections be included, or does chirality matter?
• Does preprocessing choose an arbitrary frame?
• Are generated coordinates evaluated for physical plausibility?
• Is the split unit appropriate for the claimed generalization?
• Are train and deployment coordinate sources compatible?

Related

• Equivariance
• Equivariant GNN
• Coordinate frame
• Distance geometry
• Graph construction
• Template leakage
• Structure-based modeling