  We improve the accuracy of Guidance & Control Networks (G&CNETs), trained to
represent the optimal control policies of a time-optimal transfer and a
mass-optimal landing, respectively. In both cases we leverage the dynamics of
the spacecraft, described by Ordinary Differential Equations which incorporate
a neural network on their right-hand side (Neural ODEs). Since the neural
dynamics is differentiable, the ODEs sensitivities to the network parameters
can be computed using the variational equations, thereby allowing to update the
G&CNET parameters based on the observed dynamics. We start with a
straightforward regression task, training the G&CNETs on datasets of optimal
trajectories using behavioural cloning. These networks are then refined using
the Neural ODE sensitivities by minimizing the error between the final states
and the target states. We demonstrate that for the orbital transfer, the final
error to the target can be reduced by 99% on a single trajectory and by 70% on
a batch of 500 trajectories. For the landing problem the reduction in error is
around 98-99% (position) and 40-44% (velocity). This step significantly
enhances the accuracy of G&CNETs, which instills greater confidence in their
reliability for operational use. We also compare our results to the popular
Dataset Aggregation method (DaGGER) and allude to the strengths and weaknesses
of both methods.
