  We develop methods to learn the correlation potential for a time-dependent
Kohn-Sham (TDKS) system in one spatial dimension. We start from a
low-dimensional two-electron system for which we can numerically solve the
time-dependent Schr\"odinger equation; this yields electron densities suitable
for training models of the correlation potential. We frame the learning problem
as one of optimizing a least-squares objective subject to the constraint that
the dynamics obey the TDKS equation. Applying adjoints, we develop efficient
methods to compute gradients and thereby learn models of the correlation
potential. Our results show that it is possible to learn values of the
correlation potential such that the resulting electron densities match ground
truth densities. We also show how to learn correlation potential functionals
with memory, demonstrating one such model that yields reasonable results for
trajectories outside the training set.
