Training physics‐based machine‐learning parameterizations with gradient‐free ensemble Kalman methods

Most machine learning applications in Earth system modeling currently rely on gradient-based supervised learning. This imposes stringent constraints on the nature of the data used for training (typically, residual time tendencies are needed), and it complicates learning about the interactions between machine-learned parameterizations and other components of an Earth system model. Approaching learning about process-based parameterizations as an inverse problem resolves many of these issues, since it allows parameterizations to be trained with partial observations or statistics that directly relate to quantities of interest in long-term climate projections. Here, we demonstrate the effectiveness of Kalman inversion methods in treating learning about parameterizations as an inverse problem. We consider two different algorithms: unscented and ensemble Kalman inversion. Both methods involve highly parallelizable forward model evaluations, converge exponentially fast, and do not require gradient computations. In addition, unscented Kalman inversion provides a measure of parameter uncertainty. We illustrate how training parameterizations can be posed as a regularized inverse problem and solved by ensemble Kalman methods through the calibration of an eddy-diffusivity mass-flux scheme for subgrid-scale turbulence and convection, using data generated by large-eddy simulations. We find the algorithms amenable to batching strategies, robust to noise and model failures, and efficient in the calibration of hybrid parameterizations that can include empirical closures and neural networks. Plain Language Summary Artificial intelligence represents an exciting opportunity in Earth system modeling, but its application brings its own set of challenges. One of the challenges is to train machine learning systems within Earth system models from partial or indirect data. Here, we present algorithms, known as ensemble Kalman methods, which can be used to train such systems. We demonstrate their use in situations where the data used for training are noisy, only indirectly informative about the model to be trained, and may only become available sequentially. As an example, we present training results for a state-of-the-art model for turbulence, convection, and clouds for use within Earth system models. This model is shown to learn efficiently from data in a variety of configurations, including situations where the model contains neural networks.