Spatial-Quantile Conformal Bands for Neural Operators

1. Integrate the method after a trained neural operator, such as an FNO or DeepONet, at inference time. The model must output a field hat u on the target grid. Add a positive uncertainty head hat e=softplus(h)+epsilon, or use a validation-derived residual scale map; conformal calibration itself is not backpropagated. 2. Pseudocode: for every held-out calibration pair (a_i,u_i), compute predictions hat u_i and scales hat e_i; calculate r_ij=||u_ij-hat u_ij||_2/(hat e_ij+epsilon); sort r_ij over spatial cells and find the weighted empirical quantile containing mass 1-gamma, producing S_i. Sort the S_i values, set k=min(n,ceil((n+1)(1-alpha))), and set q=S_(k). For a test input, return center hat u_j and radius q*hat e_j at each cell. 3. Compute spatial weights from uniform cells or quadrature weights normalized to sum one. Estimate the scale field from the uncertainty head or training-only residual statistics, and apply identical positivity, clipping, and interpolation rules during calibration and testing. 4. First experiment: train a small FNO on 2-D Darcy flow with disjoint training, calibration, and test sets. Compare against max-residual conformal calibration, pointwise split conformal, and Bonferroni intervals using identical model weights, calibration size, grid, and inference cost; evaluate alpha=0.1 and gamma in {0.01,0.05,0.10}. 5. Pre-register tests: repeated test batches must achieve domain coverage at least 1-gamma in at least 1-alpha of batches, allowing tolerance 0.02; at matched coverage, average band width should be at least 20% smaller than max-residual calibration; increasing gamma must monotonically decrease mean radius; and increasing calibration size must reduce coverage shortfall below 0.02. Measure weighted spatial hit fractions, not pointwise hit rates. The decisive ablation replaces the spatial-quantile score with max_j r_ij. Falsify the transfer if coverage is systematically below target, spatial quantiles do not tighten bands at matched coverage, or nonuniform quadrature weights break calibration.