Recursive Nonlocal Edge Feedback GNN

1. Integration point: replace the edge MLP in a standard message-passing GNN or graph neural ODE with two modules. The first is a recursive edge-context module g_phi; the second is the local edge-flow network f_theta. Keep the physical or observed graph sparse and do not add all-pairs edges. Initialize z_j to zero or to an edge-type embedding. 2. Pseudocode: at rollout step k, compute a pooled global summary r from node states; update each edge context with z_j = GRU(z_j, [x_tail, x_head, r]); compute u_j = MLP([x_tail, x_head, z_j, r]); compute P_j = EdgeMLP([x_tail, x_head, u_j]); aggregate q = B P; update x = NodeMLP([x, q]). Train with one-step loss plus a multi-step rollout loss. Estimate the largest Jacobian singular value or spectral radius by power iteration and optionally penalize values above one. 3. Computed from the paper: endpoint-conditioned edge flow and recursive state-to-input feedback. Estimated empirically: the feedback network, the rollout Jacobian, and the useful recursion depth. 4. First cheap experiment: use a sparse ring of coupled oscillators whose ground truth includes a distant coupling. Compare a local GNN, a dense GNN, and this model at equal parameter count. The predicted signature is a sharp rollout-stability boundary near rho(J_F) = 1: models below one should have bounded or geometrically decaying perturbations, while models above one should show exponential long-horizon divergence. At fixed sparse edge count, nonlocal-feedback error should approach dense-GNN error without proportional edge-count growth.