  Inverse problems for accelerated MRI typically incorporate domain-specific
knowledge about the forward encoding operator in a regularized reconstruction
framework. Recently physics-driven deep learning (DL) methods have been
proposed to use neural networks for data-driven regularization. These methods
unroll iterative optimization algorithms to solve the inverse problem objective
function, by alternating between domain-specific data consistency and
data-driven regularization via neural networks. The whole unrolled network is
then trained end-to-end to learn the parameters of the network. Due to
simplicity of data consistency updates with gradient descent steps, proximal
gradient descent (PGD) is a common approach to unroll physics-driven DL
reconstruction methods. However, PGD methods have slow convergence rates,
necessitating a higher number of unrolled iterations, leading to memory issues
in training and slower reconstruction times in testing. Inspired by efficient
variants of PGD methods that use a history of the previous iterates, we propose
a history-cognizant unrolling of the optimization algorithm with dense
connections across iterations for improved performance. In our approach, the
gradient descent steps are calculated at a trainable combination of the outputs
of all the previous regularization units. We also apply this idea to unrolling
variable splitting methods with quadratic relaxation. Our results in
reconstruction of the fastMRI knee dataset show that the proposed
history-cognizant approach reduces residual aliasing artifacts compared to its
conventional unrolled counterpart without requiring extra computational power
or increasing reconstruction time.
