  Model-based deep learning methods such as loop unrolling (LU) and 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. Our
approach offers a unified solution that is less parameter-sensitive, requires
no additional data, and enables simultaneous fitting of the forward model and
reconstruction in a single pass, benefiting both linear and nonlinear inverse
problems. 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. Code is available at
\texttt{https://github.com/InvProbs/A-adaptive-model-based-methods}.
