# Deep learning for inverse problems with unknown operator

arXiv: https://arxiv.org/abs/2108.02744 · HTML: https://mlsanity.com/paper/2108.02744/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 51 references that are in the map.
- First posted on arXiv 5 Aug 2021.

## Builds on (most cited first, 11 of 11)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [Nonlinear Approximation and (Deep) ReLU Networks](https://mlsanity.com/paper/1905.02199/) (2019)
- [Interpolation and Denoising of Seismic Data using Convolutional Neural Networks](https://mlsanity.com/paper/1901.07927/) (2019)
- [Limited Angle Tomography for Transmission X-Ray Microscopy Using Deep Learning](https://mlsanity.com/paper/2001.02469/) (2020)
- [Bayesian inverse problems with unknown operators](https://mlsanity.com/paper/1801.09894/) (2018)

## Built on by (most cited first, 2 of 2)

- [Med-Real2Sim: Non-Invasive Medical Digital Twins using Physics-Informed Self-Supervised Learning](https://mlsanity.com/paper/2403.00177/) (2024)
- [Adaptive Physics-Guided Neural Network](https://mlsanity.com/paper/2411.10064/) (2024)

## Abstract (arXiv)

We consider ill-posed inverse problems where the forward operator T is unknown, and instead we have access to training data consisting of functions fi and their noisy images Tfi. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of "many" training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to "fit" the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.