Adversarial negative-Sobolev PINN residual

(1) Replace the interior strong-form PINN loss, which requires evaluating div(|grad u_theta|^{p-2} grad u_theta), with the weak residual above. Use an MLP u_theta(x) for the solution and a second MLP psi_eta(x) for test functions. Enforce psi_eta = 0 on the boundary by setting psi_eta(x) = b(x) times tilde_psi_eta(x), where b(x) is a known boundary-distance factor or a product of coordinate factors on a box. (2) At each outer update, draw interior points x_i and boundary points z_j. Compute a_i = |grad u_theta(x_i)|^{p-2} grad u_theta(x_i), then estimate B with B_hat = |Omega|/N times the sum over i of [a_i dot grad psi_eta(x_i) - f(x_i) psi_eta(x_i)]. Estimate the denominator with D_hat = (|Omega|/N times the sum over i of [|psi_eta(x_i)|^p + |grad psi_eta(x_i)|^p])^{1/p} + epsilon. Ascend eta on |B_hat|/D_hat for 3 to 10 critic steps, then descend theta on L_dual + lambda_B L_B. Stop gradients through the critic during the solution-network update if joint optimization becomes unstable. (3) Estimate L_B from the fractional boundary formula by sampling random boundary pairs (z_j, z_k), using the pairwise term |g_theta(z_j)-g_theta(z_k)|^p divided by |z_j-z_k|^{m+sp}. The dual-norm supremum is estimated empirically by the critic; the exponent relation p' = p/(p-1) and the weak-form construction come directly from the paper. Clip critic gradients and optionally normalize its output after each critic update so that D_hat remains near one. (4) First test on a two-dimensional unit-square manufactured problem with p = 3 and both smooth and low-regularity target solutions. Compare against a four-layer tanh strong-form PINN and a weak Galerkin-style baseline at equal collocation counts. Track relative W^{1,p} error, fractional boundary error, dense-grid PDE residual, seed variance, and wall-clock convergence. Success means lower variance and lower W^{1,p} error at equal training steps, particularly for targets with sharp gradients or insufficient second-derivative regularity.