Inverse Problems with Learned Forward Operators

TL;DR

This paper reviews two paradigms for solving inverse problems with learned forward operators, focusing on data-driven subspace restriction and physics-based model correction, emphasizing the importance of training data for both approaches.

Contribution

It introduces and compares two novel frameworks for inverse problems using learned forward operators, highlighting their theoretical foundations and practical advantages.

Findings

Both methods benefit from training data for the forward operator and its adjoint.

The frameworks are effective in reducing computational costs while maintaining reconstruction quality.

Numerical comparisons demonstrate the advantages of each approach.

Abstract

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.