  Physics-guided deep learning is an important prevalent research topic in
scientific machine learning, which has tremendous potential in various complex
applications including science and engineering. In these applications, data is
expensive to acquire and high accuracy is required for making decisions. In
this work, we introduce an efficient physics-guided deep learning framework for
the variational modeling of nonlinear inverse problems, which is then applied
to solve an electrical impedance tomography (EIT) inverse problem. The
framework is achieved by unrolling the proposed Anderson accelerated
Gauss-Newton (GNAA) algorithm into an end-to-end deep learning method. Firstly,
we show the convergence of the GNAA algorithm in both cases: Anderson depth is
equal to one and Anderson depth is greater than one. Then, we propose three
types of strategies by combining the complementary strengths of GNAA and deep
learning: GNAA of learned regularization (GNAA-LRNet), where the singular
values of the regularization matrix are learned by a deep neural network; GNAA
of learned proximity (GNAA-LPNet), where the regularization proximal operator
is learned by using a deep neural network; GNAA of plug-and-play method
(GNAA-PnPNet) where the regularization proximal operator is replaced by a
pre-trained deep denoisers. Lastly, we present some numerical experiments to
illustrate that the proposed approaches greatly improve the convergence rate
and the quality of inverse solutions.
