Systems like aircraft and spacecraft are expensive to operate in the real world. The design, validation, and testing for such systems therefore relies on a combination of mathematical modeling, abundant numerical simulations, and a relatively small set of real-world experiments. Due to modeling errors, simplifications, and uncertainties, the data synthesized by simulation models often does not match data from the system's real-world operation. We consider the broad research question of whether this model mismatch can be significantly reduced by generative artificial intelligence models (GAIMs). Unlike text- or image-processing, where generative models have attained recent successes, GAIM development for aerospace engineering applications must not only train with scarce operational data, but their outputs must also satisfy governing equations based on natural laws, e.g., conservation laws. The scope of this paper primarily focuses on two case studies of optimally controlled systems that are commonly understood and employed in aircraft guidance, namely: minimum-time navigation in a wind field and minimum-exposure navigation in a threat field. We report GAIMs that are trained with a relatively small set, of the order of a few hundred, of examples and with underlying governing equations. By focusing on optimally controlled systems, we formulate training loss functions based on invariance of the Hamiltonian function along system trajectories. We investigate three GAIM architectures, namely: the generative adversarial network (GAN) and two variants of the variational autoencoder (VAE). We provide architectural details and thorough performance analyses of these models. The main finding is that our new models, especially the VAE-based models, are able to synthesize data that satisfy the governing equations and are statistically similar to the training data despite small volumes of training data.