With their continued increase in coverage and quality, data collected from personal air quality monitors has become an increasingly valuable tool to complement existing public health monitoring system over urban areas. However, the potential of using such `citizen science data' for automatic early warning systems is hampered by the lack of models able to capture the high-resolution, nonlinear spatio-temporal features stemming from local emission sources such as traffic, residential heating and commercial activities. In this work, we propose a machine learning approach to forecast high-frequency spatial fields which has two distinctive advantages from standard neural network methods in time: 1) sparsity of the neural network via a spike-and-slab prior, and 2) a small parametric space. The introduction of stochastic neural networks generates additional uncertainty, and in this work we propose a fast approach for forecast calibration, both marginal and spatial. We focus on assessing exposure to urban air pollution in San Francisco, and our results suggest an improvement of 35.7% in the mean squared error over standard time series approach with a calibrated forecast for up to 5 days.