Reduced-order representations of particle size distributions (PSDs) must balance reconstruction accuracy with physical compatibility if they are ultimately intended for use in atmospheric models. Linear representations based on moments or non-negative weighted integrals (NWIs) are naturally compatible with the additive structure of advection and diffusion, but their finite-dimensional form may smooth or omit localized spectral structures. Fully nonlinear features provide greater reconstruction flexibility, although they lack an explicit linear transport structure. To address this trade-off, we propose PRN, a physically guided residual framework that combines an NWI-based linear spectral backbone with a low-dimensional nonlinear correction. PRN first reconstructs a baseline PSD using N_feat−1 NWI features. A residual encoder–decoder then compresses the logarithmic difference between the target and baseline spectra into one additional feature and uses it to correct the NWI reconstruction. MOM, NWI, nonlinear feature (NLF), and PRN representations are evaluated using a common autoencoder architecture and a publicly released 33-bin liquid-droplet PSD data set derived from ensemble large-eddy simulations with spectral bin microphysics. Across ten independently trained realizations, PRN produces lower total and bin-wise reconstruction errors than MOM and NWI and substantially improves the recovery of local structures in complex spectra. The improvements are most evident for small-to-medium droplet sizes, spectral peaks, and the cloud-to-rain transition range. PRN also achieves normalized spectral-shape errors close to those of NLF, whereas its NWI component retains reproducible feature clusters across random initializations. The results show that combining a structured linear baseline with a targeted nonlinear correction provides an effective intermediate representation between purely linear and fully nonlinear PSD compression. The present study evaluates PRN offline; extending the residual feature to transport and microphysical process-rate prediction remains a subject for future investigation.

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