# Inverse Problems with Learned Forward Operators

arXiv: https://arxiv.org/abs/2311.12528 · HTML: https://mlsanity.com/paper/2311.12528/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 68 references that are in the map.
- First posted on arXiv 21 Nov 2023.

## Builds on (most cited first, 17 of 17)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Model based learning for accelerated, limited-view 3D photoacoustic tomography](https://mlsanity.com/paper/1708.09832/) (2017)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Meta-learning Pseudo-differential Operators with Deep Neural Networks](https://mlsanity.com/paper/1906.06782/) (2019)
- [Approximate k-space models and Deep Learning for fast photoacoustic reconstruction](https://mlsanity.com/paper/1807.03191/) (2018)
- [Model-corrected learned primal-dual models for fast limited-view photoacoustic tomography](https://mlsanity.com/paper/2304.01963/) (2023)
- [Joint Activity Detection and Channel Estimation for Clustered Massive Machine Type Communications](https://mlsanity.com/paper/2305.02935/) (2023)

## Built on by (most cited first, 4 of 4)

- [Physics-informed neural networks (PINNs) for numerical model error approximation and superresolution](https://mlsanity.com/paper/2411.09728/) (2024)
- [Bayesian Model Parameter Learning in Linear Inverse Problems: Application in EEG Focal Source Imaging](https://mlsanity.com/paper/2501.13109/) (2025)
- [Evaluating structural uncertainty in accelerated MRI: are voxelwise measures useful surrogates?](https://mlsanity.com/paper/2503.10527/) (2025)
- [Uncertainty-quantified Pulse Signal Recovery from Facial Video using Regularized Stochastic Interpolants](https://mlsanity.com/paper/2604.10777/) (2026)

## Abstract (arXiv)

Solving inverse problems requires the knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants that do not compromise the reconstruction quality are desired. This chapter reviews reconstruction methods in inverse problems with learned forward operators that follow two different paradigms. The first one is completely agnostic to the forward operator and learns its restriction to the subspace spanned by the training data. The framework of regularisation by projection is then used to find a reconstruction. The second one uses a simplified model of the physics of the measurement process and only relies on the training data to learn a model correction. We present the theory of these two approaches and compare them numerically. A common theme emerges: both methods require, or at least benefit from, training data not only for the forward operator, but also for its adjoint.