Physics-Constrained Reduced-Order Modeling of Collision-Coalescence with Advectable Embeddings: Monotonic Mass Partition Scheme

Cloud and precipitation processes remain major sources of uncertainty in atmospheric models, partly because traditional bulk microphysics represent particle size distributions (PSDs) using only a few moments, limiting accuracy in key processes such as collision–coalescence. Machine learning (ML) emulators offer a promising path forward by learning microphysical rates from detailed bin schemes, but their performance depends critically on the choice of latent space embedding. Recent work has proposed Non-negative Weighted Integrals (NWIs) as advectable, linear embeddings of the PSD. However, the lack of physical interpretability of NWI embeddings makes it difficult to design stable and accurate ML-based rate models. To address this, we introduce two complementary contributions: a new advectable embedding framework called Monotonic Mass Partition (MMP), and a physics-informed process rate model architecture. MMP embeddings impose monotonic constraints on the underlying weight functions of NWIs so that the droplet mass follows a single-direction transfer along the components during collision-coalescence process. The process rate model exploits this structure by predicting only positive rate factors that explicitly embeds the quadratic nature of collision-coalescence. Using bin microphysics as a benchmark, we trained reduced-order models (ROMs) based on MMP embeddings and traditional moments to emulate the collision–coalescence process. All ROMs produce stable latent-space evolution, with ML-based ROMs outperforming the traditional power-law autoconversion and accretion parameterization, and MMP embeddings achieving further gains with the most accurate prediction of precipitation onset. Our results demonstrate that embedding-level physical constraints are crucial and highlight the potential of advectable embeddings as next-generation prognostic representations for ML-based microphysics parameterizations.