Solving Inverse Problems with Model Mismatch using Untrained Neural
  Networks within Model-based Architectures
Peimeng Guan Naveed Iqbal Mark A. Davenport Mudassir Masood
Model-based deep learning methods such as \emph{loop unrolling} (LU) and
\emph{deep equilibrium model} (DEQ) extensions offer outstanding performance in
solving inverse problems (IP). These methods unroll the optimization iterations
into a sequence of neural networks that in effect learn a regularization
function from data. While these architectures are currently state-of-the-art in
numerous applications, their success heavily relies on the accuracy of the
forward model. This assumption can be limiting in many physical applications
due to model simplifications or uncertainties in the apparatus. To address
forward model mismatch, we introduce an untrained forward model residual block
within the model-based architecture to match the data consistency in the
measurement domain for each instance. We propose two variants in well-known
model-based architectures (LU and DEQ) and prove convergence under mild
conditions. The experiments show significant quality improvement in removing
artifacts and preserving details across three distinct applications,
encompassing both linear and nonlinear inverse problems. Moreover, we highlight
reconstruction effectiveness in intermediate steps and showcase robustness to
random initialization of the residual block and a higher number of iterations
during evaluation.