  Predicting the spatiotemporal variation in streamflow along with uncertainty
quantification enables decision-making for sustainable management of scarce
water resources. Process-based hydrological models (aka physics-based models)
are based on physical laws, but using simplifying assumptions which can lead to
poor accuracy. Data-driven approaches offer a powerful alternative, but they
require large amount of training data and tend to produce predictions that are
inconsistent with physical laws. This paper studies a constrained reasoning and
learning (CRL) approach where physical laws represented as logical constraints
are integrated as a layer in the deep neural network. To address small data
setting, we develop a theoretically-grounded training approach to improve the
generalization accuracy of deep models. For uncertainty quantification, we
combine the synergistic strengths of Gaussian processes (GPs) and deep temporal
models (i.e., deep models for time-series forecasting) by passing the learned
latent representation as input to a standard distance-based kernel. Experiments
on multiple real-world datasets demonstrate the effectiveness of both CRL and
GP with deep kernel approaches over strong baseline methods.
