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。

水资源管理的带不确定性量化的径流预测：一种约束推理与学习方法

Streamflow Prediction with Uncertainty Quantification for Water  Management: A Constrained Reasoning and Learning Approach

Regional rainfall-runoff modeling is an old but still mostly out-standing
problem in Hydrological Sciences. The problem currently is that traditional
hydrological models degrade significantly in performance when calibrated for
multiple basins together instead of for a single basin alone. In this paper, we
propose a novel, data-driven approach using Long Short-Term Memory networks
(LSTMs), and demonstrate that under a 'big data' paradigm, this is not
necessarily the case. By training a single LSTM model on 531 basins from the
CAMELS data set using meteorological time series data and static catchment
attributes, we were able to significantly improve performance compared to a set
of several different hydrological benchmark models. Our proposed approach not
only significantly outperforms hydrological models that were calibrated
regionally but also achieves better performance than hydrological models that
were calibrated for each basin individually. Furthermore, we propose an
adaption to the standard LSTM architecture, which we call an Entity-Aware-LSTM
(EA-LSTM), that allows for learning, and embedding as a feature layer in a deep
learning model, catchment similarities. We show that this learned catchment
similarity corresponds well with what we would expect from prior hydrological
understanding.

使用气象时间序列数据和静态流域属性，我们提出了一种基于数据驱动的方法，在使用 LSTMs 单个模型对 531 个流域进行训练的情况下，显著提高了性能，比符合地区性条件的水文模型和针对单个流域进行校准的水文模型的表现更好，同时还提出了 Entity-Aware-LSTM 的改进网络架构使学习、嵌入和特征层深度学习可以在其上实现，同时提出的这个背景与之前的水文理解相符。