Representing cloud microphysical processes in large scale atmospheric models is challenging because many processes depend on the details of the droplet size distribution (DSD, the spectrum of droplets with different sizes in a cloud). While full or partial statistical moments of droplet size distributions are the typical basis set used in bulk models, prognostic moments are limited in their ability to represent microphysical processes across the range of conditions experienced in the atmosphere. Microphysical parameterizations employing prognostic moments are known to suffer from structural uncertainty in their representations of inherently higher dimensional cloud processes, which limit model fidelity and lead to forecasting errors. Here we investigate how data-driven reduced order modeling can be used to learn predictors for microphysical process rates in bulk microphysics schemes in an unsupervised manner from higher dimensional bin distributions, by simultaneously learning lower dimensional representations of droplet size distributions and predicting the evolution of the microphysical state of the system. Droplet collision-coalescence, the main process for generating warm rain, is estimated to have an intrinsic dimension of 3. This intrinsic dimension provides a lower limit on the number of degrees of freedom needed to accurately represent collision-coalescence in models. We demonstrate how deep learning based reduced-order modeling can be used to discover intrinsic coordinates describing the microphysical state of the system, where process rates such as collision-coalescence are globally linearized. These implicitly learned representations of the DSD retain more information about the DSD than typical moment-based representations.