# Deep Neural Networks and Inverse Problems

## Prerequisites

- Functional analysis, in particular, operator theory
- Deep learning for image reconstruction
- Basic knowledge of PyTorch or Tensorflow

## Description

The goal of inverse problems is to recover model parameters from a set of indirect measurements, for example, recovering images from data. In most experimental sciences, inverse problems play an important role since the physical observations are always mediated by measuring devices. An inverse problem is mathematically formulated as an operator inversion problem where the operator inverts the model's measurements in the absence of noise. The operator is referred to as "forward operator and is typically not invertible, making the associated inverse problem "ill-posed". Solving the ill-posed inverse problem by simply minimizing the miss-fit against the data leads to overfitting. To overcome this challenge, one needs to introduce apriori information to obtain solutions that are consistent with the physics of the measured signals. This can be, for example, the fact that the measured signals are sparse. Apriori information can be either modeled by first principles or learned from data; combining these approaches has yielded the best performance.

## References

- The Analysis of Linear Partial Differential Operators IV
- (https://link.springer.com/book/10.1007/978-3-642-00136-9)
- Solving ill-posed inverse problems using iterative deep neural networks
- (https://arxiv.org/abs/1704.04058)
- Task adapted reconstruction for inverse problems (https://iopscience.iop.org/article/10.1088/1361-6420/ac28ec/meta)