  Data-driven algorithms, in particular neural networks, can emulate the effect
of sub-grid scale processes in coarse-resolution climate models if trained on
high-resolution climate simulations. However, they may violate key physical
constraints and lack the ability to generalize outside of their training set.
Here, we show that physical constraints can be enforced in neural networks,
either approximately by adapting the loss function or to within machine
precision by adapting the architecture. As these physical constraints are
insufficient to guarantee generalizability, we additionally propose to
physically rescale the training and validation data to improve the ability of
neural networks to generalize to unseen climates.
