Unlike the traditional subgrid scale parameterizations used in climate models, current machine learning (ML) parameterizations are only tuned offline, by minimizing a loss function on outputs from high resolution models. This approach often leads to numerical instabilities and long-term biases. Here, we propose a method to design tunable ML parameterizations and calibrate them online. The calibration of the ML parameterization is achieved in two steps. First, some model parameters are included within the ML model input. This ML model is fitted at once for a range of values of the parameters, using an offline metric. Second, once the ML parameterization has been plugged into the climate model, the parameters included among the ML inputs are optimized with respect to an online metric quantifying errors on long-term statistics. We illustrate our method with a simple dynamical system. Our approach significantly reduces long-term biases of the ML model.

The development of atmospheric parameterizations based on neural networks is often hampered by numerical instability issues. Previous attempts to replicate these issues in a toy model have proven ineffective. We introduce a new toy model for atmospheric dynamics, which consists in an extension of the Lorenz'63 model to a higher dimension. While neural networks trained on a single orbit can easily reproduce the dynamics of the Lorenz'63 model, they fail to reproduce the dynamics of the new toy model, leading to unstable trajectories. Instabilities become more frequent as the dimension of the new model increases, but are found to occur even in very low dimension. Training the neural network on a different learning sample, based on Latin Hypercube Sampling, solves the instability issue. Our results suggest that the design of the learning sample can significantly influence the stability of dynamical systems driven by neural networks.