  Neural closure models have recently been proposed as a method for efficiently
approximating small scales in multiscale systems with neural networks. The
choice of loss function and associated training procedure has a large effect on
the accuracy and stability of the resulting neural closure model. In this work,
we systematically compare three distinct procedures: "derivative fitting",
"trajectory fitting" with discretise-then-optimise, and "trajectory fitting"
with optimise-then-discretise. Derivative fitting is conceptually the simplest
and computationally the most efficient approach and is found to perform
reasonably well on one of the test problems (Kuramoto-Sivashinsky) but poorly
on the other (Burgers). Trajectory fitting is computationally more expensive
but is more robust and is therefore the preferred approach. Of the two
trajectory fitting procedures, the discretise-then-optimise approach produces
more accurate models than the optimise-then-discretise approach. While the
optimise-then-discretise approach can still produce accurate models, care must
be taken in choosing the length of the trajectories used for training, in order
to train the models on long-term behaviour while still producing reasonably
accurate gradients during training. Two existing theorems are interpreted in a
novel way that gives insight into the long-term accuracy of a neural closure
model based on how accurate it is in the short term.
