  We consider the Earth-Venus mass-optimal interplanetary transfer of a
low-thrust spacecraft and show how the optimal guidance can be represented by
deep networks in a large portion of the state space and to a high degree of
accuracy. Imitation (supervised) learning of optimal examples is used as a
network training paradigm. The resulting models are suitable for an on-board,
real-time, implementation of the optimal guidance and control system of the
spacecraft and are called G&CNETs. A new general methodology called Backward
Generation of Optimal Examples is introduced and shown to be able to
efficiently create all the optimal state action pairs necessary to train
G&CNETs without solving optimal control problems. With respect to previous
works, we are able to produce datasets containing a few orders of magnitude
more optimal trajectories and obtain network performances compatible with real
missions requirements. Several schemes able to train representations of either
the optimal policy (thrust profile) or the value function (optimal mass) are
proposed and tested. We find that both policy learning and value function
learning successfully and accurately learn the optimal thrust and that a
spacecraft employing the learned thrust is able to reach the target conditions
orbit spending only 2 permil more propellant than in the corresponding
mathematically optimal transfer. Moreover, the optimal propellant mass can be
predicted (in case of value function learning) within an error well within 1%.
All G&CNETs produced are tested during simulations of interplanetary transfers
with respect to their ability to reach the target conditions optimally starting
from nominal and off-nominal conditions.
