  Fast and accurate solution of time-dependent partial differential equations
(PDEs) is of key interest in many research fields including physics,
engineering, and biology. Generally, implicit schemes are preferred over the
explicit ones for better stability and correctness. The existing implicit
schemes are usually iterative and employ a general-purpose solver which may be
sub-optimal for a specific class of PDEs. In this paper, we propose a neural
solver to learn an optimal iterative scheme for a class of PDEs, in a
data-driven fashion. We attain this objective by modifying an iteration of an
existing semi-implicit solver using a deep neural network. Further, we prove
theoretically that our approach preserves the correctness and convergence
guarantees provided by the existing iterative-solvers. We also demonstrate that
our model generalizes to a different parameter setting than the one seen during
training and achieves faster convergence compared to the semi-implicit schemes.
