Equivariance and Invariance in Graph Networks

Symmetry is a modeling asset; building permutation, and where relevant geometric, symmetry into GNNs cuts data needs and enables breeder-to-burner transfer.

Two symmetries that matter

A correct model of the sensor graph should not depend on the arbitrary order in which sensors are listed (permutation symmetry), and physical vector quantities should transform consistently under rotation/reflection (geometric symmetry). Encoding these directly means the network cannot waste capacity learning them from data, and it generalizes across configurations it has not seen.

Permutation equivariance / invariance:

  For any node permutation P:
    GNN_layer(P x, P A P^T) = P * GNN_layer(x, A)   (equivariant)
    readout( P x ) = readout( x )                   (invariant)

  Guaranteed if aggregation AGG is order-independent
  (sum / mean / attention) -> node ordering irrelevant

Why it enables transfer

Permutation equivariance is what lets a GNN trained on the breeder's diagnostic topology apply to the burner's different sensor set, and lets sensors be added or retired mid-life without retraining. The learned message and update functions act on features and local structure, not on fixed node indices, so the model transfers where the physics is analogous.

# invariant graph readout (order-independent)
def readout(h_nodes):
    return concat(sum(h_nodes, axis=0),
                  mean(h_nodes, axis=0),
                  max(h_nodes, axis=0))   # all permutation-invariant

Geometric equivariance where needed

For quantities that are vectors in space - field perturbations, displacements - the stack uses equivariant message functions so that rotating the inputs rotates the outputs identically. This keeps learned dynamics physically consistent and reduces the data needed to cover orientations, which matters given how little device-relevant data exists, especially for the burner.

• Permutation equivariance: node order irrelevant; enables transfer.
• Robust to sensor add/remove without retraining from scratch.
• Geometric equivariance: vector fields transform consistently.
• Symmetry priors cut data demand and improve generalization.

Symmetry is treated as free, verified physics knowledge: building it in is preferred over hoping the network learns it, particularly in the data-poor burner regime where every prior counts.