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

Hang-Cheng Dong, Pengcheng Cheng, Shuhuan Li

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 ($O(δ^{0.5})$ 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.

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

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 ( for ), 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 () when trained on coarse grids () and tested on significantly finer grids (up to ). The proposed method bridges the gap between rigorous regularization theory and data-driven operator learning.
Paper Structure (28 sections, 4 theorems, 38 equations, 3 figures)

This paper contains 28 sections, 4 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Operator Framework
  3. The Forward Problem
  4. Spectral Decomposition and Ill-posedness
  5. Regularization as Spectral Filtering
  6. Limitations of Classical Filters:
  7. Motivation for our Approach:
  8. Methodology: Learnable Spectral Filtering
  9. The SC-Net Architecture
  10. Spectral Projection Layer (Analysis)
  11. Learnable Spectral Filter Block
  12. Network Structure:
  13. Crucial Constraint:
  14. Spectral Synthesis Layer (Synthesis)
  15. Loss Function with Sobolev Regularization
  16. ...and 13 more sections

Key Result

Proposition 2.1

The operator $\mathcal{K}: L^2(\Omega) \to L^2(\partial \Omega)$ is a linear compact operator.

Figures (3)

  • Figure 1: Convergence analysis of the relative $L^2$ reconstruction error with respect to noise level $\delta$ (Log-Log Scale).
  • Figure 2: Visualization of the learned spectral filter profile (Red) compared to Oracle Tikhonov (Blue) and the ideal Truncated SVD (Black Dotted) under noise level $\delta=5\%$.
  • Figure 3: Zero-shot generalization to unseen discretization resolutions. The model trained on $N=256$ is directly applied to $N=512$, $1024$, and $2048$.

Theorems & Definitions (10)

  • Proposition 2.1: Compactness of $\mathcal{K}$
  • proof
  • Definition 2.1: Ill-posedness
  • Theorem 4.1: Stability of SC-Net
  • proof
  • Definition 4.1: Oracle Filter
  • Theorem 4.2: Approximation Capability
  • proof
  • Theorem 4.3: Total Error Bound
  • proof