Harmonic Global Latent Channels

(1) Exact integration point: use this as the output head of a mesh neural operator or graph neural network whose output is a discrete differential k-form, such as edge values for k = 1 or face fluxes for k = 2. Precompute the mesh incidence matrices B_k and B_{k+1} and mass matrices M_k and M_{k-1}. Compute a basis H for the discrete harmonic subspace by solving the nullspace constraints B_{k+1}H = 0 and M_{k-1}^{-1}B_k^T M_k H = 0, then M_k-orthonormalize the columns. (2) Pseudocode: z = GNN(features, mesh); u_raw = local_head(z); coeff = H.T @ M_k @ u_raw; u = u_raw - H @ coeff; a = global_pool(z, cycle_features); y = u + H @ a; loss = task_loss(y, target) + lambda_period * norm(P @ y - period_target)^2 + lambda_gauge * norm(H.T @ M_k @ u)^2. For forecasting, concatenate the current global coefficient a_t to every local node or edge feature and predict a_{t+1}. (3) Compute H, the weighted projection H(H^T M_k input), and P exactly from the mesh; estimate only the neural mappings from input features to the local field and coefficients. If period labels are unavailable, use P @ target as the target period. For changing meshes, recompute H and P or transport coefficients through a consistent homology-cycle basis, fixing basis signs and ordering to prevent label permutations. (4) First cheap experiment: use an annulus or square-with-hole mesh, generate Poisson or Maxwell fields with randomized circulation, and compare a four-layer message-passing baseline with this harmonic-head model at equal parameter count. Measure field L2 error, period error, and performance after mesh refinement. Success means lower circulation error at the same depth, particularly when the hole is farther apart than the baseline receptive field, plus improved zero-shot transfer to refined meshes.