Abstract
A promising method for improving the representation of clouds in climate
models, and hence climate projections, is to develop machine
learning-based parameterizations using output from global
storm-resolving models. While neural networks can achieve
state-of-the-art performance, they are typically climate model-specific,
require post-hoc tools for interpretation, and struggle to predict
outside of their training distribution. To avoid these limitations, we
combine symbolic regression, sequential feature selection, and physical
constraints in a hierarchical modeling framework. This framework allows
us to discover new equations diagnosing cloud cover from coarse-grained
variables of global storm-resolving model simulations. These analytical
equations are interpretable by construction and easily transferable to
other grids or climate models. Our best equation balances performance
and complexity, achieving a performance comparable to that of neural
networks ($R^2=0.94$) while remaining simple (with only 13
trainable parameters). It reproduces cloud cover distributions more
accurately than the Xu-Randall scheme across all cloud regimes
(Hellinger distances $<0.09$), and matches neural networks
in condensate-rich regimes. When applied and fine-tuned to the ERA5
reanalysis, the equation exhibits superior transferability to new data
compared to all other optimal cloud cover schemes. Our findings
demonstrate the effectiveness of symbolic regression in discovering
interpretable, physically-consistent, and nonlinear equations to
parameterize cloud cover.