<p>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.</p>