  Excited-state dynamics simulations are a powerful tool to investigate
photo-induced reactions of molecules and materials and provide complementary
information to experiments. Since the applicability of these simulation
techniques is limited by the costs of the underlying electronic structure
calculations, we develop and assess different machine learning models for this
task. The machine learning models are trained on {\emph ab initio} calculations
for excited electronic states, using the methylenimmonium cation
(CH$_2$NH$_2^+$) as a model system. For the prediction of excited-state
properties, multiple outputs are desirable, which is straightforward with
neural networks but less explored with kernel ridge regression. We overcome
this challenge for kernel ridge regression in the case of energy predictions by
encoding the electronic states explicitly in the inputs, in addition to the
molecular representation. We adopt this strategy also for our neural networks
for comparison. Such a state encoding enables not only kernel ridge regression
with multiple outputs but leads also to more accurate machine learning models
for state-specific properties. An important goal for excited-state machine
learning models is their use in dynamics simulations, which needs not only
state-specific information but also couplings, i.e., properties involving pairs
of states. Accordingly, we investigate the performance of different models for
such coupling elements. Furthermore, we explore how combining all properties in
a single neural network affects the accuracy. As an ultimate test for our
machine learning models, we carry out excited-state dynamics simulations based
on the predicted energies, forces and couplings and, thus, show the scopes and
possibilities of machine learning for the treatment of electronically excited
states.
