Physical Information, Numerical Resolution, and Learned Approximation in Born Scattering

Learning wave-scattering operators presents three main challenges: resolving transmitted information, controlling discretization error, and approximating the operator from data. Using quadrature-weighted finite-aperture Born operators and DeepONets, we analyze their interplay. Operator spectra across frequency bands show a plateau and a steep tail, with a shifting knee that complicates global power-law fitting. Effective rank identifies key transmission directions. Receiver refinement shows first- or second-order convergence (rectangle or trapezoidal quadrature), making receiver requirements dependent on the integration rule and accuracy, while aperture extension affects physical coverage. In two-band studies, linear branches outperform MLPs; with width 64, wavelength matching cuts high-band linear-model errors by 66% and 52%. Refitting coefficients in learned bases remove most in-distribution error, but training-based POD with ridge prediction is more accurate at equal data, revealing an approximation gap. Full-wave checks find Born discrepancies of 4.5%–10.4% at nominal contrast, quantifying model accuracy but not neural full-wave generalization. Overall, we clarify the roles of physics, numerics, and learning, showing accuracy is distribution-dependent and not universally set by effective rank.