Sparse Multiscale Kernel-Frame Operator

(1) Integrate at the representation boundary of a neural operator: take sampled coordinates X and field values v in R^M, convert v into multilevel coefficient blocks alpha_0,...,alpha_J, process those blocks with an MLP or graph neural network, and decode the predicted blocks at output coordinates Y. Construct C_0=X and obtain C_1,...,C_J by farthest-point sampling, voxel subsampling, or selecting every 2^j-th grid point. If boundary coverage is poor, add auxiliary centers G_j near the boundary. Set rho_j to a multiple of the median nearest-neighbor distance in Xi_j; begin with rho_j=2.5 times that distance. Use a compactly supported Wendland function, for example phi(t)=(1-t)_+^4(4t+1), so each row of A_j has only local nonzeros. (2) Pseudocode: initialize q=0; for j=0,...,J, compute r=v-q, form sparse A_j with A_j[i,k]=phi(||x_i-xi_jk||/rho_j), solve (A_j^T A_j+lambda_j I)alpha_j=A_j^T r using sparse conjugate gradients, then update q=q+A_j alpha_j. Concatenate alpha_j with a learned level embedding and feed them to the coefficient network, producing beta_j. For query points Y, form sparse A_j(Y) and return v_hat(Y)=sum_j A_j(Y)beta_j. (3) Compute directly from the construction: kernel matrices, sparse linear solves, residuals, and per-level reconstructions. Estimate lambda_j by validation, and log the condition number or conjugate-gradient iteration count of each regularized Gram matrix. (4) First experiment: train on a 2-D Darcy or elliptic multiscale coefficient-to-solution dataset using 64x64 fields. Compare against an FNO and U-Net with matched parameter count and training steps. Test relative L2 error, high-frequency residual error, irregular-point-cloud generalization, sparse-kernel memory, and wall-clock time. The method is successful if it lowers test error at equal parameter count or reaches baseline error with fewer active coefficients, while retaining stable reconstruction when the output coordinates are changed.