Inverse Problems with Learned Forward Operators

TL;DR

This work analyzes two data-driven paradigms for inverse problems with learned forward operators: (i) regularisation by projection, which learns the forward operator restricted to a training subspace and uses projection-based minimisation, and (ii) data-driven model correction, which learns a correction to a cheaper approximate forward model. It develops the mathematical framework for both approaches, including convergence results for injective and non-injective cases, and extends to explicit forward–adjoint corrections with convergence guarantees under gradient-alignment conditions. The theory highlights that successful reconstructions benefit from training data not only for the forward operator but also for its adjoint, a recurring theme across sections. The chapter substantiates these ideas with PAT experiments, showing that learned forward/normal operators perform well under projection-based regularisation, while explicit model corrections can yield strong improvements when trained adequately (including trajectory-aware strategies). Overall, the work clarifies how data-driven corrections and subspace projections can yield computationally efficient yet accurate reconstructions, and it points to practical training strategies that leverage adjoint information and manifold-based approaches for scalability.

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.

Figures (6)

• Figure 1: Illustration of the forward operator $A$ and the approximate model $\widetilde{A}$, as well as corresponding normal operators. The sensor is indicated with a red line (left/right images), in the measurements (middle) the red line corresponds to initial time $t=0$.
• Figure 2: Projected variational regularisation. Experiments with $n=1500$ training samples. Both learned forward operator and learned normal operator perform well, the reconstructions with the learned normal operator being perhaps a bit sharper. Surprisingly, combining the learned forward model with an approximate one $\widetilde{A}$ decreases the reconstruction quality.
• Figure 3: Projected variational regularisation. The effect of the number of training samples. The method is surprisingly robust with respect to the number of samples. Reconstructions are still reasonable for as few as $500$ samples (top row), and even for $100$ (middle row) samples the location of the disc is identified reasonably well. If we decrease the number of samples further so that the support of the samples does not overlap with the support of the ground truth then the location of the disc is identified incorrectly (bottom row).
• Figure 4: Comparison of reconstructions using the approximate model and learned corrections under various training schemes. (Top row) Ground truth and reconstruction with the accurate model, the sensor is located at the top. (Second row) Reconstruction using the approximate operator $\widetilde{A}$ and reconstruction using the approximation error method. (Third row) Learned correction of the forward model trained on disks only (left) and recursively on the trajectory (right). (Bottom row) Forward-adjoint correction trained on disks only (left) and recursively on the trajectory (right).
• Figure 5: (Left) Alignment \ref{['eqn:AligenementVariational']} of the approximate gradient and the gradient of the accurate data term $A^*(A {x}^{(k)} - y)$ for different training approaches. (Right) True data term $\|A{x}^{(k)} - y\|_{\mathcal{Y}}$ for different methods, tracked throughout the gradient descent scheme.

Theorems & Definitions (9)

• Theorem 1: asp-kor-sch-2020
• Remark 1
• Theorem 2: asp-kor-sch-2020
• Theorem 3: engl:1996
• Theorem 4: asp-kor-sch-2020
• Theorem 5: asp-kor-sch-2020, slightly modified
• Definition 1: generalised Bregman distance
• Theorem 6: asp-kor-sch-2020, slightly modified
• Theorem 7: Convergence to a neighbourhood of the accurate solution lunz21