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.

预测流域水量的时空变化及不确定性量化有助于可持续管理宝贵的水资源，本文研究了约束推理和学习方法，在深度神经网络中将物理定律作为逻辑约束层进行集成，并结合高斯过程和深度时间模型进行不确定性量化。实验证明，这些方法在多个真实数据集上比基线方法具有更好的效果。