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.
Abstract (translated)
求解病态反问题需要有效的正则化策略，以稳定反演过程对测量噪声的鲁棒性。尽管经典方法（如吉洪诺夫正则化）依赖启发式参数调优，而标准深度学习方法常缺乏可解释性及跨分辨率泛化能力，我们提出了SC-Net（谱校正网络），一种新颖的算子学习框架。SC-Net在前向算子的谱域中运行，学习逐点自适应滤波函数，依据信噪比重新加权谱系数。我们提供理论分析表明SC-Net逼近连续反演算子，保证离散化不变性。在一维积分方程上的数值实验证明SC-Net：（1）达到理论极小极大最优收敛速率（当s=p=1.5时为O(δ^0.5)），匹配理论下界；（2）学习可解释的锐截止滤波器，性能超越最优吉洪诺夫正则化；（3）展现零样本超分辨率能力，在粗网格（N=256）上训练后，于显著更细网格（最高N=2048）测试时仍保持稳定重建误差（约0.23）。该方法弥合了严格正则化理论与数据驱动算子学习之间的差距。
URL
https://arxiv.org/abs/2603.20602
