  Data assimilation is the process of fusing information from imperfect
computer simulations with noisy, sparse measurements of reality to obtain
improved estimates of the state or parameters of a dynamical system of
interest. The data assimilation procedures used in many geoscience
applications, such as numerical weather forecasting, are variants of the
our-dimensional variational (4D-Var) algorithm. The cost of solving the
underlying 4D-Var optimization problem is dominated by the cost of repeated
forward and adjoint model runs. This motivates substituting the evaluations of
the physical model and its adjoint by fast, approximate surrogate models.
Neural networks offer a promising approach for the data-driven creation of
surrogate models. The accuracy of the surrogate 4D-Var solution depends on the
accuracy with each the surrogate captures both the forward and the adjoint
model dynamics. We formulate and analyze several approaches to incorporate
adjoint information into the construction of neural network surrogates. The
resulting networks are tested on unseen data and in a sequential data
assimilation problem using the Lorenz-63 system. Surrogates constructed using
adjoint information demonstrate superior performance on the 4D-Var data
assimilation problem compared to a standard neural network surrogate that uses
only forward dynamics information.
