1. Species distribution models (SDMs) can be used to predict how individual species---and whole assemblages of species---will respond to a changing environment. Until now, these models have either assumed (1) that species' occurrence probabilities are uncorrelated, or (2) that species respond linearly to preselected environmental variables. These two assumptions currently prevent ecologists from modeling assemblages with realistic co-occurrence and species richness properties. 2. This paper introduces a stochastic feedforward neural network, called mistnet, which makes neither assumption. Thus, unlike most SDMs, mistnet can account for non-independent co-occurrence patterns driven by unobserved environmental heterogeneity. And unlike recently proposed Joint SDMs, mistnet can also learn nonlinear functions relating species' occurrence probabilities to environmental predictors. 3. Mistnet makes more accurate predictions about the North American bird communities found along Breeding Bird Survey transects than several alternative methods tested. In particular, typical assemblages held out of sample for validation were nearly 50,000 times more likely under the mistnet model than under independent combinations of single-species models. 4. Apart from improved accuracy, mistnet shows two other important benefits for ecological research and management. First: by analyzing co-occurrence data, mistnet can identify unmeasured---and perhaps unanticipated---environmental variables that drive species turnover. For example, mistnet identified a strong grassland/forest gradient, even though only temperature and precipitation were given as model inputs. Second: mistnet is able to take advantage of incomplete data sets to guide its predictions towards more realistic assemblages. For example, mistnet automatically adjusts its expectations to include more forest-associated species in response to a stray observation of a forest-dwelling warbler.