  We present an approach to construct approximate Koopman-type decompositions
for dynamical systems depending on static or time-varying parameters. Our
method simultaneously constructs an invariant subspace and a parametric family
of projected Koopman operators acting on this subspace. We parametrize both the
projected Koopman operator family and the dictionary that spans the invariant
subspace by neural networks and jointly train them with trajectory data. We
show theoretically the validity of our approach, and demonstrate via numerical
experiments that it exhibits significant improvements over existing methods in
solving prediction problems, especially those with large state or parameter
dimensions, and those possessing strongly non-linear dynamics. Moreover, our
method enables data-driven solution of optimal control problems involving
non-linear dynamics, with interesting implications on controllability.
