We develop a convolutional neural network (CNN) model to predict the diffusivity of cations in nanoparticle-based electrolytes, and use it to identify the characteristics of morphologies which exhibit optimal transport properties. The ground truth data is obtained from kinetic Monte Carlo (kMC) simulations of cation transport parameterized using a multiscale modeling strategy. We implement deep learning approaches to quantitatively link the diffusivity of cations to the spatial arrangement of the nanoparticles. We then integrate the trained CNN model with a topology optimization algorithm for accelerated discovery of nanoparticle morphologies that exhibit optimal cation diffusivities at a specified nanoparticle loading, and we investigate the ability of the CNN model to quantitatively account for the influence of interparticle spatial correlations on cation diffusivity. Finally, using data-driven approaches, we explore how simple descriptors of nanoparticle morphology correlate with cation diffusivity, thus providing a physical rationale for the observed optimal microstructures. The results of this study highlight the capability of CNNs to serve as surrogate models for structure--property relationships in composites with monodisperse spherical particles, which can in turn be used with inverse methods to discover morphologies that produce optimal target properties.