Interpretable Operator Learning for Inverse Problems via Adaptive Spectral Filtering: Convergence and Discretization Invariance

TL;DR

This paper introduces SC-Net, an operator learning framework that uses spectral domain filtering to solve inverse problems with theoretical guarantees, interpretability, and resolution invariance, outperforming classical regularization methods.

Contribution

SC-Net is a novel spectral operator learning method that guarantees discretization invariance and achieves minimax optimal convergence rates for inverse problems.

Findings

Achieves minimax optimal convergence rate ($O( ext{delta}^{0.5})$)

Learns interpretable sharp-cutoff spectral filters

Maintains stable reconstructions across different grid resolutions

Abstract

Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and standard deep learning approaches often lack interpretability and generalization across resolutions, we propose SC-Net (Spectral Correction Network), a novel operator learning framework. SC-Net operates in the spectral domain of the forward operator, learning a pointwise adaptive filter function that reweights spectral coefficients based on the signal-to-noise ratio. We provide a theoretical analysis showing that SC-Net approximates the continuous inverse operator, guaranteeing discretization invariance. Numerical experiments on 1D integral equations demonstrate that SC-Net: (1) achieves the theoretical minimax optimal convergence rate ($O(\delta^{0.5})$ for $s=p=1.5$), matching theoretical lower bounds; (2) learns interpretable sharp-cutoff filters that outperform Oracle Tikhonov regularization; and (3) exhibits zero-shot super-resolution, maintaining stable reconstruction errors ($\approx 0.23$) when trained on coarse grids ($N=256$) and tested on significantly finer grids (up to $N=2048$). The proposed method bridges the gap between rigorous regularization theory and data-driven operator learning.