We study the problem of learning a function that maps context observations (input) to parameters of a submodular function (output). Our motivating case study is a specific type of vehicle routing problem, in which a team of Unmanned Ground Vehicles (UGVs) can serve as mobile charging stations to recharge a team of Unmanned Ground Vehicles (UAVs) that execute persistent monitoring tasks. {We want to learn the mapping from observations of UAV task routes and wind field to the parameters of a submodular objective function, which describes the distribution of landing positions of the UAVs .} Traditionally, such a learning problem is solved independently as a prediction phase without considering the downstream task optimization phase. However, the loss function used in prediction may be misaligned with our final goal, i.e., a good routing decision. Good performance in the isolated prediction phase does not necessarily lead to good decisions in the downstream routing task. In this paper, we propose a framework that incorporates task optimization as a differentiable layer in the prediction phase. Our framework allows end-to-end training of the prediction model without using engineered intermediate loss that is targeted only at the prediction performance. In the proposed framework, task optimization (submodular maximization) is made differentiable by introducing stochastic perturbations into deterministic algorithms (i.e., stochastic smoothing). We demonstrate the efficacy of the proposed framework using synthetic data. Experimental results of the mobile charging station routing problem show that the proposed framework can result in better routing decisions, e.g. the average number of UAVs recharged increases, compared to the prediction-optimization separate approach.