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 for replacing detailed binned yet computationally expensive schemes with efficient emulations. However, many existing emulations still rely on moments as prognostic variables, inheriting structural limitations from traditional bulk schemes. In contrast, latent variables directly discovered by ML are potential to represent PSDs more accurately, but their inherent nonlinearity breaks the conservation property under advection and diffusion, limiting their applicability in online simulations. To address this dilemma, we propose Weighted Integral Parameters (WIPs), formulated as weighted integrals of PSD with learnable weight functions, providing the most general mathematical form for advectable microphysical prognostic variables. Using autoencoders that are physics-informed by WIP’s formulation to learn the optimal PSD representations, we conducted unsupervised learning over a liquid droplet PSD dataset generated from ensemble large eddy simulations with Spectral Bin Microphysics, to compare WIPs with traditional moment approaches in bulk schemes on their ability to represent “actual” PSDs. Results show that WIPs can automatically capture the critical information of medium-sized droplets unprecedentedly with traditional moment approaches, and outperform partial and full integral moments in terms of PSD reconstruction error, indicating superior PSD information compression efficiency. With these properties, WIPs are potential to replace moments as fully prognostic variables to build more accurate ML-based bin-emulating schemes.