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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 has 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 time step to time step. This maintains the physical understanding built into the models, while 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 initialized 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.