Graph Neural Networks: Message-Passing Mathematics

The diagnostic constellation and the coupled subsystems are graphs; message passing is the mathematics that lets a network reason over their topology.

Fusion state as a graph

Many objects in the stack are naturally graphs: the 60+ port diagnostic constellation (nodes = sensors, edges = physical/flux-surface proximity), and the coupled plant subsystems (nodes = magnet, plasma, blanket, DEC train). A graph neural network (GNN) processes these by passing messages along edges, so a node's updated state reflects its neighborhood - exactly the locality physics has.

Message-passing layer (one round):

  m_ij = phi_msg( h_i, h_j, e_ij )        (message on edge i<-j)
  a_i  = AGG_{j in N(i)} m_ij             (permutation-invariant)
  h_i' = phi_upd( h_i, a_i )              (node update)

  h_i : node feature ,  e_ij : edge feature (geometry, coupling)
  AGG : sum / mean / attention (invariant to neighbor order)

Why the aggregator must be invariant

Physical neighbors have no intrinsic order, so the aggregation must be permutation-invariant - sum, mean, max, or attention-weighted sum. This invariance is what lets the same GNN handle a diagnostic being added or retired without retraining from scratch, a practical requirement across a machine's life and across the breeder-to-burner transfer.

# one message-passing round over the diagnostic graph
for i in nodes:
    msgs = [phi_msg(h[i], h[j], e[i,j]) for j in neigh(i)]
    a_i  = attention_sum(msgs)        # invariant aggregation
    h[i] = gru(h[i], a_i)             # gated update keeps temporal state

Depth equals reach

Each message-passing round extends a node's receptive field by one hop; K rounds let information travel K edges. Too few rounds miss long-range coupling; too many cause over-smoothing, where node features collapse together. The stack tunes depth to the physical correlation length of the graph - short for local sensor imputation, longer for whole-machine coupling.

• Nodes: sensors, or subsystems, with feature vectors.
• Edges: physical adjacency, flux-surface links, coupling strength.
• Rounds K: receptive field radius; balance reach vs over-smoothing.
• Aggregation: permutation-invariant, robust to node add/remove.

Message passing is the shared substrate for the stack's two GNN roles - imputing dropped diagnostics and modeling coupled-subsystem dynamics - each covered on its own page.