发表机构
Deutscher Wetterdienst（德国气象局）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将通用常微分方程（UODEs）框架应用于暖雨形成参数化，仅用预报状态变量训练，得到的闭合关系可准确复现KCE轨迹，还能改进现有暖雨参数化，兼具替代模型与解析参数化优化工具的作用。
AI 中文摘要
暖雨形成的参数化是云微物理学中的长期难题，因为体云变量的演化无法唯一从动力学收集方程（KCE）推导得出。通用常微分方程（UODEs）是结合神经网络与常微分方程的科学机器学习框架，可直接从时间序列学习动力学算子。本研究将UODE框架应用于暖雨形成，通过训练近似KCE的超滴模型生成的轨迹完成学习。与多数机器学习方法不同，该方法仅使用预报状态变量进行训练，无需将自动转化或碰并率作为目标。学习得到的闭合关系在训练域内可准确复现KCE轨迹，且虽未受明确约束，其产生的自动转化和碰并率与KCE诊断结果高度相似。分析学习到的算子可为广泛使用的Seifert和Beheng（2001）暖雨参数化的结构提供新见解，尤其表明碰并的经验抑制主要补偿了小雨分数下自动转化的高估。基于这些发现，本研究提出了一种改进的解析公式，在保留原参数化简洁性的同时提升了与KCE的一致性。结果表明，UODEs不仅可作为精确的替代模型，还可作为理解和改进云微物理过程解析参数化的工具。
英文摘要
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.
