  The modelling of small-scale processes is a major source of error in climate
models, hindering the accuracy of low-cost models which must approximate such
processes through parameterization. Red noise is essential to many operational
parameterization schemes, helping model temporal correlations. We show how to
build on the successes of red noise by combining the known benefits of
stochasticity with machine learning. This is done using a physically-informed
recurrent neural network within a probabilistic framework. Our model is
competitive and often superior to both a bespoke baseline and an existing
probabilistic machine learning approach (GAN) when applied to the Lorenz 96
atmospheric simulation. This is due to its superior ability to model temporal
patterns compared to standard first-order autoregressive schemes. It also
generalises to unseen scenarios. We evaluate across a number of metrics from
the literature, and also discuss the benefits of using the probabilistic metric
of hold-out likelihood.
