Universal ordinary differential equations and the parameterization of warm-rain processes
The parameterization of warm-rain formation is a long-standing problem in cloud microphysics because the evolution of bulk cloud variables cannot be derived uniquely from the kinetic collection equation (KCE). Universal ordinary differential equations (UODEs)provide a scientific machine learning framework that combines neural networks with ordinary differential equations and can learn dynamical operators directly from time series. Here, the UODE framework is applied to warm-rain formation by training on trajectories generated with a super-droplet model that approximates the KCE. In contrast to most previous machine-learning approaches, only the prognostic state variables are used for training, without requiring autoconversion or accretion rates as targets. The learned closure accurately reproduces the KCE trajectories within the training domain and, although not explicitly constrained to do so, yields autoconversion and accretion rates that closely resemble those diagnosed from the KCE. Analyzing the learned operator provides new insight into the structure of the widely used Seifert and Beheng (2001) warm-rain parameterization. In particular, it suggests that the empirical suppression of accretion primarily compensates for an overestimation of autoconversion at small rain fractions. Motivated by these findings, a refined analytical formulation is proposed that improves agreement with the KCE while retaining the simplicity of the original parameterization. The results demonstrate that UODEs can serve not only as accurate surrogate models but also as a tool for understanding and improving analytical parameterizations of cloud microphysical processes.