  Integration of machine learning (ML) models of unresolved dynamics into
numerical simulations of fluid dynamics has been demonstrated to improve the
accuracy of coarse resolution simulations. However, when trained in a purely
offline mode, integrating ML models into the numerical scheme can lead to
instabilities. In the context of a 2D, quasi-geostrophic turbulent system, we
demonstrate that including an additional network in the loss function, which
emulates the state of the system into the future, produces offline-trained ML
models that capture important subgrid processes, with improved stability
properties.
