Structure-Factor Coordinates for Finite-Aperture Operator Learning

Learning wave responses from aperture geometry requires resolving interference between spatially distributed elements. We introduce a raster-free physical interface that represents this interference through point-set Fourier intensity and multiscale angular moments of the structure factor. In paired acoustic controls, the interface reduces response MAE by 22.7% for Transolver-MS and 21.4% for Point-DeepONet. When the square-triangle family is entirely withheld, the gains persist across three seeds: 10.2-18.2% for Transolver-MS and 18.2-20.5% for Point-DeepONet across two training settings. SURFNO complements the same interface with a fitted kernel transform and learned residual, and performs competitively with spectral baselines. Antenna and scalar diffraction benchmarks reuse the Fourier-intensity construction under different observation measures, while solver-verified inverse design separates representation quality from feasible mask-space search. Equal-spectrum counterexamples delineate the information discarded by spectral compression. Together, the results establish structure-factor coordinates as a reusable physical interface across predictor architectures and an unseen aperture family; finite-window observations and geometry-dependent coupling require complementary geometric or source-state information.