Interpretable operator learning for inverse problems via adaptive spectral filtering: Convergence and resolution...

We study a learned spectral regularization method for compact linear inverse problems. SC-Net (Spectral Correction Network) applies a bounded neural damping factor to each retained pseudoinverse coefficient, conditioned on the observed coefficient and singular value. At fixed truncation, we prove local stability and approximation of continuous damping profiles. Under a Hilbert-scale source condition and algebraic singular-value decay, a uniformly bounded network family satisfying an approximation-to-identity condition attains the classical truncated-spectral convergence order. The estimate does not require a noise-uniform network Lipschitz constant. We also establish discretization consistency for a fixed reconstruction rule under compatible spectral and projection assumptions. Synthetic one-dimensional tests exhibit scaling consistent with the predicted order. Conditioning on the observed coefficient reduces mean error by 12.5% relative to a singular-value-only ablation. SC-Net outperforms the tested classical baselines at low and moderate noise; a discrepancy-principle-anchored Fourier neural operator (FNO-1D) has the lowest error among implementable methods on the fixed-operator benchmark. Tests with Laplace and correlated Gaussian noise assess transfer without retraining. For two-dimensional Gaussian deblurring, Tikhonov-anchored SC-Net has lower errors than an equal-information operator-conditioned FNO at the tested in-range and held-out blur widths. This comparison shows an accuracy advantage but does not isolate sensitivity to operator shift. A fixed-mode experiment shows decreasing cross-resolution discrepancy; its grid-dependent conditioning limits the comparison with the fixed-rule theorem.