Ensembles of geophysical models improve prediction accuracy and express uncertainties. We develop a novel data-driven ensembling strategy for combining geophysical models using Bayesian Neural Networks, which infers spatiotemporally varying model weights and bias, while accounting for heteroscedastic uncertainties in the observations. This produces more accurate and uncertainty-aware predictions without sacrificing interpretability. Applied to the prediction of total column ozone from an ensemble of 15 chemistry-climate models, we find that the Bayesian neural network ensemble (BayNNE) outperforms existing methods for ensembling physical models, achieving a 49.4% reduction in RMSE for temporal extrapolation, and a 67.4% reduction in RMSE for polar data voids, compared to a weighted mean. Uncertainty is also well-characterized, with 91.9% of the data points in our extrapolation validation dataset lying within 2 standard deviations and 98.9% within 3 standard deviations.