Smooth-Plus-Boundary-Lifting Network

Use two coordinate networks with shared low-level features: N_int_theta(x) for the smooth branch and N_bd_theta(x) for the boundary quotient. Define u_theta(x)=N_int_theta(x)+d_delta(x)^a N_bd_theta(x), using a smoothed distance d_delta when needed, and set u_theta=0 outside the domain. At every training step, sample ordinary interior points X and oversample a boundary strip X_b={x:d(x)<rho}. Evaluate the combined solution, compute r=P[u_theta]-f, and minimize the PDE residual plus a branch regularizer. The exact paper ingredient is the direct-sum structure H_q^(2a+s) plus d^a times a lifting. In code, approximate smoothness with automatic-differentiation gradients evaluated on X, and approximate boundary-branch localization with m_b(x)=exp(-d(x)/rho). Use L_sep=mean(||nabla N_int||^2)+lambda_bd mean(m_b||d_delta^(1-a)nabla N_bd||^2)+lambda_far mean((1-m_b)N_bd^2). Initialize the interior branch with low-frequency Fourier features and the boundary branch with ordinary positional features. Test on a unit disk or square using manufactured solutions containing both a smooth term and d^a times a smooth term. Compare with a single unconstrained PINN and the single factored architecture, keeping parameter count and collocation budget fixed. Track total L2 error, boundary-strip error, residual convergence, and branch magnitudes away from the boundary. Success means lower boundary-layer error at equal parameter count without worsening interior accuracy.