Inverse-Laplacian Residual Loss

(1) Integration point: modify the PDE-loss branch of a PINN training step, not the network architecture. For a batch of fixed interior grid points, evaluate $u_\theta$, use automatic differentiation to compute the PDE residual vector $r_\theta=[\mathcal{N}(u_\theta)(x_i)-f(x_i)]_{i=1}^n$, and replace its MSE by a discrete inverse-Laplacian quadratic form. Dirichlet data should initially be imposed exactly with $u_\theta(x)=g_{\mathrm{ext}}(x)+d(x)v_\theta(x)$, where $d$ vanishes on the boundary; this avoids mixing an unpreconditioned boundary MSE with the filtered interior loss. (2) Pseudocode: precompute a finite-difference or finite-element mass matrix $M$ and Dirichlet stiffness matrix $K$ on the collocation grid; at each step compute r = pde_residual(net, x_grid), solve z = solve(K, M @ r), set loss_pde = r.T @ M @ z, add any data or initial-condition losses, backpropagate, and update Adam or LBFGS. Use a Cholesky factorization of $K$ for small fixed grids, or conjugate gradients with a fixed tolerance for larger grids. Detach neither z nor the solve output if gradients through the loss are desired; alternatively use the symmetric quadratic-form gradient $2J_r^\top M K^{-1}M r$. (3) The paper supplies the $H^{-1}$ construction and its elliptic smoothing rationale. The engineer estimates $M$ and $K$ numerically from the domain and chooses the grid resolution; verify the solve by checking $\|Kz-Mr\|/\|Mr\|$. Normalize by the number of grid points and optionally add a small ridge $K+\varepsilon M$ only if the linear solve is poorly conditioned. (4) First experiment: solve 2D Poisson $-\Delta u=f$ on $[0,1]^2$ with nonzero Dirichlet data using the same MLP, collocation budget, optimizer, and random seeds for standard residual MSE versus $H^{-1}$ loss. Compare relative $L^2$ and $H^1$ solution errors, gradient-norm variance, residual-spectrum decay, wall-clock time, and iterations to a fixed error. Success means faster loss descent or fewer optimizer steps at equal PDE evaluations, reduced sensitivity to the interior/boundary scale ratio, and lower $H^1$ error; a useful ablation compares exact $K^{-1}$ against one or three approximate CG iterations.