Inverse-inequality resolution control

1. Integration point: modify the derivative-residual loss and collocation scheduler in a PINN, neural operator, or coordinate MLP. The relevant quantities are feature-parameter separation \(\underline h\), derivative order \(r\) used by the PDE residual, and a lower-order diagnostic norm of order \(s\). 2. Pseudocode: initialize feature parameters \(\Theta\); compute \(\underline h=\min_{i\ne j}\|\theta_i-\theta_j\|\) after identifying antipodal duplicates; choose the largest candidate derivative order \(r<k+1/2\) satisfying \(\underline h^{-(r-s)}\le C_{\max}\). At each training phase, estimate lower-order residual norm \(E_s\) on a validation point set, compute the bound proxy \(B_r=\underline h^{-(r-s)}E_s\), and either use derivative loss \(L_r\) only when \(B_r\) is below a threshold or refine by adding well-separated features and collocation points. A practical loss is \(L=L_s+\lambda_rL_r\) with \(\lambda_r=\min(1,\tau/B_r)\). 3. The theorem supplies the exponent \(r-s\) and admissible range \(r<k+1/2\); the constant, actual Sobolev norms, and domain transfer must be estimated empirically. For nonspherical inputs, replace \(\underline h\) by minimum hidden-weight separation or a measured feature-kernel length scale. 4. First experiment: train width-128 and width-256 coordinate ReLU MLPs on a 2D Poisson or biharmonic problem with first-, second-, and fourth-order residual penalties. Compare fixed derivative losses against the adaptive rule at equal FLOPs. Track gradient-norm variance, residual-Jacobian condition number, dense-grid PDE residual, and final solution error. Success is that the adaptive rule prevents exploding derivative gradients and achieves equal or lower dense-grid error without increasing the collocation budget. A falsifying result is no reduction in instability or no correlation between predicted factor \(\underline h^{-(r-s)}\) and measured derivative amplification.