# Solving Inverse Problems With Deep Neural Networks -- Robustness Included?

arXiv: https://arxiv.org/abs/2011.04268 · HTML: https://mlsanity.com/paper/2011.04268/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Nov 2020.

## Built on by (most cited first, 30 of 58)

- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Solving the Inverse Problem of Electrocardiography for Cardiac Digital Twins: A Survey](https://mlsanity.com/paper/2406.11445/) (2024)
- [Near-Exact Recovery for Tomographic Inverse Problems via Deep Learning](https://mlsanity.com/paper/2206.07050/) (2022)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Adversarial Robustness of MR Image Reconstruction under Realistic Perturbations](https://mlsanity.com/paper/2208.03161/) (2022)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [Connections between Numerical Algorithms for PDEs and Neural Networks](https://mlsanity.com/paper/2107.14742/) (2021)
- [To be or not to be stable, that is the question: understanding neural networks for inverse problems](https://mlsanity.com/paper/2211.13692/) (2022)
- [On the Robustness of Normalizing Flows for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.04319/) (2022)
- [Learning Provably Robust Estimators for Inverse Problems via Jittering](https://mlsanity.com/paper/2307.12822/) (2023)
- [Memory-efficient model-based deep learning with convergence and robustness guarantees](https://mlsanity.com/paper/2206.04797/) (2022)
- [Stability of Image-Reconstruction Algorithms](https://mlsanity.com/paper/2206.07128/) (2022)
- [On Adversarial Robustness of Deep Image Deblurring](https://mlsanity.com/paper/2210.02502/) (2022)
- [Evaluating Similitude and Robustness of Deep Image Denoising Models via Adversarial Attack](https://mlsanity.com/paper/2306.16050/) (2023)
- [Glimpse: Generalized Locality for Scalable and Robust CT](https://mlsanity.com/paper/2401.00816/) (2024)
- [NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problems](https://mlsanity.com/paper/2203.00804/) (2022)
- [Data Models for Dataset Drift Controls in Machine Learning With Optical Images](https://mlsanity.com/paper/2211.02578/) (2022)
- [On Instabilities of Conventional Multi-Coil MRI Reconstruction to Small Adverserial Perturbations](https://mlsanity.com/paper/2102.13066/) (2021)
- [Subtle Data Crimes: Naively training machine learning algorithms could lead to overly-optimistic results](https://mlsanity.com/paper/2109.08237/) (2021)
- [Unveiling and Mitigating Adversarial Vulnerabilities in Iterative Optimizers](https://mlsanity.com/paper/2504.19000/) (2025)
- [AAPM DL-Sparse-View CT Challenge Submission Report: Designing an Iterative Network for Fanbeam-CT with Unknown Geometry](https://mlsanity.com/paper/2106.00280/) (2021)
- [Training Adaptive Reconstruction Networks for Blind Inverse Problems](https://mlsanity.com/paper/2202.11342/) (2022)
- [Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems](https://mlsanity.com/paper/2206.00934/) (2022)
- [Localized adversarial artifacts for compressed sensing MRI](https://mlsanity.com/paper/2206.05289/) (2022)
- [Using explainability to design physics-aware CNNs for solving subsurface inverse problems](https://mlsanity.com/paper/2211.08651/) (2022)
- [Representing Noisy Image Without Denoising](https://mlsanity.com/paper/2301.07409/) (2023)
- [Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning](https://mlsanity.com/paper/2407.13666/) (2024)
- [How many measurements are enough? Bayesian recovery in inverse problems with general distributions](https://mlsanity.com/paper/2505.10630/) (2025)

## Abstract (arXiv)

In the past five years, deep learning methods have become state-of-the-art in solving various inverse problems. Before such approaches can find application in safety-critical fields, a verification of their reliability appears mandatory. Recent works have pointed out instabilities of deep neural networks for several image reconstruction tasks. In analogy to adversarial attacks in classification, it was shown that slight distortions in the input domain may cause severe artifacts. The present article sheds new light on this concern, by conducting an extensive study of the robustness of deep-learning-based algorithms for solving underdetermined inverse problems. This covers compressed sensing with Gaussian measurements as well as image recovery from Fourier and Radon measurements, including a real-world scenario for magnetic resonance imaging (using the NYU-fastMRI dataset). Our main focus is on computing adversarial perturbations of the measurements that maximize the reconstruction error. A distinctive feature of our approach is the quantitative and qualitative comparison with total-variation minimization, which serves as a provably robust reference method. In contrast to previous findings, our results reveal that standard end-to-end network architectures are not only resilient against statistical noise, but also against adversarial perturbations. All considered networks are trained by common deep learning techniques, without sophisticated defense strategies.