Ground-state fractional regularizer

1. Integration point: use a one-dimensional coordinate MLP or PINN that predicts a scalar field on x\in[-1,1], with a known singular or interface location shifted to x=0. Replace the final output by u_ heta(x)=h_{2,\beta,s}(x)v_ heta(x), where \beta is estimated from boundary asymptotics or selected from a small grid and s=\exp(a) is either fixed or learned. Add the regularizer \mathcal R_{\alpha,\beta,s} to the task loss; the first double integral is evaluated on sampled point pairs. 2. Pseudocode: sample collocation points x_i and pairs (x_i,y_i); compute h_i=(|x_i|^\beta(1_{x_i>0}+s1_{x_i<0}))^{1/2}; compute v_i=v_ heta(x_i), u_i=h_i v_i; set E_pair=mean[h_i h_j (v_i-v_j)^2 |x_i-y_j|^{-\alpha-1}]; set E_hardy=mean[u_i^2 |x_i|^{-\alpha}]; optimize L=L_task+\lambda_f\mathcal A_\alpha E_pair/2+\lambda_h E_hardy. Exclude identical pairs and clip |x-y| below a small \varepsilon for finite-precision stability. 3. Compute \mathcal A_\alpha exactly from the displayed gamma-function formula. Use \gamma(\alpha,\rho) to initialize \lambda_h, with \rho=\beta/2 or a validation-selected value; do not assume the theoretical coefficient remains sharp for arbitrary data. Estimate the optional optimal asymmetry by one or two gradient steps on a=\log s, or by evaluating a small set of s values and choosing the lowest validation loss. 4. First experiment: train a four-layer width-128 tanh MLP on one-dimensional fractional Poisson data whose solution is u(x)=|x|^\beta g(x), comparing ordinary MSE plus an unweighted fractional penalty against the ground-state parameterization. Use 2,048 collocation points and 32,768 random pairs. Measure relative L2 error, boundary-error scaling, gradient variance, and convergence at equal pair-evaluation FLOPs. Success means lower error near x=0 and a smoother loss curve, ideally reaching a fixed relative error with fewer collocation samples; also test sensitivity to the chosen \beta and whether learned s improves asymmetric interface cases.