  Dynamical weather and climate prediction models underpin many studies of the
Earth system and hold the promise of being able to make robust projections of
future climate change based on physical laws. However, simulations from these
models still show many differences compared with observations. Machine learning
has been applied to solve certain prediction problems with great success, and
recently it's been proposed that this could replace the role of
physically-derived dynamical weather and climate models to give better quality
simulations. Here, instead, a framework using machine learning together with
physically-derived models is tested, in which it is learnt how to correct the
errors of the latter from timestep to timestep. This maintains the physical
understanding built into the models, whilst allowing performance improvements,
and also requires much simpler algorithms and less training data. This is
tested in the context of simulating the chaotic Lorenz '96 system, and it is
shown that the approach yields models that are stable and that give both
improved skill in initialised predictions and better long-term climate
statistics. Improvements in long-term statistics are smaller than for single
time-step tendencies, however, indicating that it would be valuable to develop
methods that target improvements on longer time scales. Future strategies for
the development of this approach and possible applications to making progress
on important scientific problems are discussed.
