Nonlinear Laplacian Equilibrium GNN Layer

Integrate this as an implicit layer in a message-passing GNN. Inputs are node features $x\in\mathbb{R}^{n\times h}$, an undirected edge list, and a balanced demand vector $d$; output node potentials $\phi$, edge flows $f$, or their concatenation with $x$. Compute an edge embedding $z_e=\mathrm{MLP}([x_u,x_v,x_u-x_v])$. Use it to parameterize a monotone edge law, for example $g_e(s)=a_e+b_es$ with $b_e\ge 0$, $w_e(s)=\rho'_e(s)=\operatorname{softplus}(g_e(s))+\epsilon$, and either a closed-form integrated law or numerical quadrature for $\rho_e(s)$. Initialize $\phi_0=0$ and fix one reference node. At iteration $t$, calculate $s=B^\top\phi_t$, residual $r=B\rho(s)-\alpha d$, and weights $w=\rho'(s)$. Every $K$ iterations, construct the sparse matrix $\widetilde J=B\operatorname{diag}(w)B^\top$; otherwise reuse the previous matrix or preconditioner. Solve $\widetilde J\Delta=-r$ with conjugate gradient plus algebraic multigrid or sparse approximate Cholesky, then use backtracking: test $\phi_t+\eta\Delta$, halve $\eta$ until the residual norm decreases, and accept the update. Stop at a residual tolerance or fixed budget. For the MVP, backpropagate through 5--10 unrolled iterations; subsequently use implicit differentiation by solving $J^\top v=\partial L/\partial\phi$. The paper supplies the residual and Jacobian structure; estimate the refresh period, damping, solver tolerance, and preconditioner quality empirically. Test first on Cora or a synthetic stochastic block model for node classification, comparing against a 6-layer GCN at equal sparse matrix-vector products. Record validation accuracy, residual norm, number of Laplacian solves, and wall-clock time. Success requires improved accuracy or robustness at equal propagation cost, fewer sequential iterations, and no divergence when learned edge slopes span several orders of magnitude.