提出可解释的谱滤波网络，实现逆问题求解的稳定与跨分辨率泛化。
Interpretable Operator Learning for Inverse Problems via Adaptive Spectral Filtering: Convergence and Discretization Invariance
- 在前向算子的谱域学习自适应滤波器，根据信噪比重加权谱系数。
- 达到理论最优收敛率（δ⁰·⁵），且在粗网格训练后零样本超分辨重建误差仅约0.23。
- 滤波器可解释，适用于需理论保障与跨尺度泛化的逆问题求解场景。
求解病态逆问题需要有效的正则化策略以抑制测量噪声。传统方法如Tikhonov正则化依赖经验调参，而标准深度学习方法缺乏可解释性且分辨率泛化能力差。本文提出SC-Net（谱校正网络），在前向算子的谱域中学习点对点自适应滤波函数，根据信噪比重加权谱系数。理论分析表明，SC-Net逼近连续逆算子，保证离散化不变性。1D积分方程上的数值实验显示：(1) 达到理论最小最大最优收敛率（s=p=1.5时为O(δ⁰·⁵)），匹配理论下界；(2) 学习到可解释的锐截止滤波器，性能优于已知最优的Oracle Tikhonov正则化；(3) 具备零样本超分辨率能力，在粗网格（N=256）训练后测试于更细网格（最高达N=2048），重建误差保持稳定（≈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鸿沟。
原文摘要 · Abstract (English)
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 ($O(δ^{0.5})$ 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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。