Stiffness-energy supervision without FEM labels

Integrate the method as the loss layer of a graph neural network or mesh-conditioned MLP. The network receives node coordinates, element connectivity, material parameters, boundary-condition masks, and loads, and outputs displacements only for free degrees of freedom. For each minibatch item, assemble the sparse free-free stiffness matrix K and construct the load vector F. Use the following computation: u_hat = net(mesh_features, material_features, loads); Ku = sparse_matvec(K, u_hat); energy = 0.5 * dot(u_hat, Ku) - dot(F, u_hat); loss = mean(energy over batch); loss.backward(); optimizer.step(). This is exactly Pi_h(u_hat), with the paper's K_ff represented by K. Autodifferentiation sends K u_hat minus F into the network output. Do not call a sparse linear solver during training. Enforce prescribed displacements by predicting free variables only, or set constrained output entries to their known values before evaluating the energy. The guarantee requires K to be symmetric positive definite, so verify sparse symmetry and exclude or regularize mechanisms with K plus epsilon times the identity. Compare energy-only training against displacement-MSE training and supervised stiffness-norm training on a small 2D elasticity dataset with exact FEM solutions retained only for evaluation. Measure stiffness-norm error, Euclidean error, energy gap, training wall-clock time, and the number of reference solves used to create labels. Success means comparable stiffness-norm accuracy with zero training-time reference solves and substantially lower dataset-generation cost; a first target is at least 5x lower total preprocessing cost at similar final stiffness-norm error.