Exploring Advectable Latent Representations for Droplet Size Distributions With Physics-Informed Autoencoders

Investigating the role of clouds and precipitation in the Earth system necessitates microphysical schemes capable of accurately describing the evolution of hydrometeor particle size distribution (PSD), while maintaining low computational costs implementable in atmospheric models. Machine learning (ML) offers a promising approach to replace computationally expensive bin microphysical schemes with efficient emulations. However, many existing ML emulations predict moments of PSDs as prognostic variables, inheriting structural limitations from traditional bulk schemes. In contrast, latent variables directly discovered by ML have the potential to represent PSDs more accurately. However, their inherent nonlinearity breaks the conservation property under advection and diffusion, limiting their applicability in online simulations. To address this dilemma, we propose Non-negative weighted integrals (NWIs), formulated as weighted integrals of PSD with learnable non-negative weight functions. NWI provides the most general mathematical form for advectable microphysical prognostic variables. We conducted unsupervised learning over a liquid droplet PSD data set generated from ensemble large eddy simulations with Spectral Bin Microphysics (SBM). We used autoencoders that are physics-informed by NWI’s formulation to learn the optimal PSD representations from the data, and compared NWIs with traditional moment approaches in bulk schemes on their ability to represent PSDs in actual bin scheme simulations. Results show that NWIs can capture the critical information of medium-sized droplets, and outperform traditional cloud-rain moment approaches in terms of PSD reconstruction error, indicating improved PSD information compression efficiency. With these properties, NWIs are advantageous over moments as fully prognostic variables to build accurate ML-based bin-emulating schemes.