Geometric observability gating

Integrate the method at the sparse-observation interface of a graph neural operator, graph diffusion model, or latent graph SSM. Let the graph be \(G=(V,E)\), with edge lengths \(\ell_{ij}\), observed node set \(\omega\), input features \(X_0\), and target states \(X(t)\). First compute \(d_i=d(i,\omega)\) using multi-source Dijkstra for weighted graphs or BFS for unweighted graphs. Normalize distances by the median edge length and cap them at a chosen percentile to avoid one disconnected component dominating the loss. Use a diffusion backbone such as \(X(t)=\exp(-tL)X_0\), approximated with \(K\) Chebyshev polynomial terms, or a learned residual update \(X_{k+1}=X_k-\Delta t\,\Phi_\theta(L,X_k)\). At every sampled time \(t\), calculate \(h_i(t)=\exp[-\kappa d_i^2/(t+\epsilon)]\), with \(\kappa\in\{0.1,0.25,0.45\}\), and optimize the normalized weighted reconstruction loss in the fourth formula. If the decoder uses observation-to-node cross-attention, add \(b_i(t)=\log h_i(t)=-\kappa d_i^2/(t+\epsilon)\) to the attention logits for queries at node \(i\), so distant nodes cannot make overconfident early reads from sensors. The paper supplies the exponential distance/time law and the critical \(1/2\) scale; distances, the graph Laplacian, diffusion times, and loss normalization are computed by the implementation. Begin with a 2D grid or stochastic-block graph, hide all but 5% of nodes as sensors, and compare a six-layer Graph Diffusion Network with uniform loss against the gated version at equal parameter count and FLOPs. Measure early-time gradient variance, reconstruction MSE versus distance to sensors, and final MSE at fixed sensor density. Success is lower gradient variance and at least 10% lower far-node reconstruction MSE without degrading near-sensor accuracy.