Exact-Curl Neural Field Output

Integrate this at the output head of a neural operator, CNN, or graph neural network that predicts a 2-D or 3-D vector field on a fixed mesh. In 2-D, make the network output a scalar potential A on grid vertices or cell corners. In 3-D, make it output a vector potential a=(a_x,a_y,a_z) on mesh edges. Apply fixed differentiable incidence stencils: B_x=(A[i,j+1]-A[i,j-1])/(2dy), B_y=-(A[i+1,j]-A[i-1,j])/(2dx) in the simple periodic 2-D case, or use a Yee-style staggered curl in 3-D. For a recurrent simulator, predict delta_A=Net(state), set A_next=A+delta_A, and compute B_next=C(A_next). Use the reconstructed B_next in data, PDE, energy, and force losses; omit the divergence penalty. Pseudocode is: state -> Net -> delta_A; A_next=A+delta_A; B_next=C(A_next); loss=L_data(B_next)+lambda*L_physics(B_next); backpropagate through C. The paper supplies the potential/curl construction and DC=0 identity. The engineer must implement compatible boundary conditions and verify DC numerically. First test a small Fourier neural operator or message-passing GNN on 2-D incompressible advection or magnetic-vortex rollouts. Compare direct-field prediction plus divergence penalty against the curl head at equal parameter count. Measure field error, maximum absolute cell divergence, and long-horizon energy drift. Success means divergence stays at floating-point or stencil-roundoff level without penalty tuning, with equal or lower rollout error and improved stability.