# Deep regularization networks for inverse problems with noisy operators

arXiv: https://arxiv.org/abs/2506.07008 · HTML: https://mlsanity.com/paper/2506.07008/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 62 references that are in the map.
- First posted on arXiv 8 Jun 2025.

## Builds on (most cited first, 19 of 19)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics](https://mlsanity.com/paper/1705.07115/) (2017)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [Learning Regularization Parameters of Inverse Problems via Deep Neural Networks](https://mlsanity.com/paper/2104.06594/) (2021)
- [Bilevel approaches for learning of variational imaging models](https://mlsanity.com/paper/1505.02120/) (2015)
- [Learning Regularization Parameter-Maps for Variational Image Reconstruction using Deep Neural Networks and Algorithm Unrolling](https://mlsanity.com/paper/2301.05888/) (2023)
- [Neural-network-based regularization methods for inverse problems in imaging](https://mlsanity.com/paper/2312.14849/) (2023)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [Bilevel Optimization, Deep Learning and Fractional Laplacian Regularization with Applications in Tomography](https://mlsanity.com/paper/1907.09605/) (2019)
- [DRIP: Deep Regularizers for Inverse Problems](https://mlsanity.com/paper/2304.00015/) (2023)
- [Deep learning for full-field ultrasonic characterization](https://mlsanity.com/paper/2301.02378/) (2023)
- [Whiteness-based bilevel learning of regularization parameters in imaging](https://mlsanity.com/paper/2403.07026/) (2024)
- [Network scaling and scale-driven loss balancing for intelligent poroelastography](https://mlsanity.com/paper/2411.08886/) (2024)

## Abstract (arXiv)

A supervised learning approach is proposed for regularization of large inverse problems where the main operator is built from noisy data. This is germane to superresolution imaging via the sampling indicators of the inverse scattering theory. We aim to accelerate the spatiotemporal regularization process for this class of inverse problems to enable real-time imaging. In this approach, a neural operator maps each pattern on the right-hand side of the scattering equation to its affiliated regularization parameter. The network is trained in two steps which entails: (1) training on low-resolution regularization maps furnished by the Morozov discrepancy principle with nonoptimal thresholds, and (2) optimizing network predictions through minimization of the Tikhonov loss function regulated by the validation loss. Step 2 allows for tailoring of the approximate maps of Step 1 toward construction of higher quality images. This approach enables direct learning from test data and dispenses with the need for a-priori knowledge of the optimal regularization maps. The network, trained on low-resolution data, quickly generates dense regularization maps for high-resolution imaging. We highlight the importance of the training loss function on the network's generalizability. In particular, we demonstrate that networks informed by the logic of discrepancy principle lead to images of higher contrast. In this case, the training process involves many-objective optimization. We propose a new method to adaptively select the appropriate loss weights during training without requiring an additional optimization process. The proposed approach is synthetically examined for imaging damage evolution in an elastic plate. The results indicate that the discrepancy-informed regularization networks not only accelerate the imaging process, but also remarkably enhance the image quality in complex environments.