Characterized by essential complexity, dynamism, and dynamics, streamflow forecasting presents a great challenge to hydrologists. Long short-term memory (LSTM) streamflow forecast model has received a lot of attention in recent years due to its powerful non-linear modeling ability. But probabilistic streamflow forecasting has rarely been addressed by the LSTM approach. In this study, a probabilistic Long Short-Term Memory network coupled with the Gaussian process (GP) is proposed to deal with the probabilistic daily streamflow forecasting. Moreover, considering that changing mean and variance over time exist in the daily streamflow time series, the heteroscedastic Gaussian process regression is adopted to produce a varying prediction interval. The proposed method encapsulates the inductive biases of the LSTM recurrent network and retains the non-parametric, probabilistic property of Gaussian processes. The performance of the proposed model is investigated by predicting the daily streamflow time series collected from the upper Yangtze River and its tributaries. Artificial neuron network, generalized linear model, heteroscedastic GP, and regular LSTM models are also developed for comparison. Results indicated that the performance of the proposed model is satisfying. It improves prediction accuracy as well as provides an adaptive prediction interval, which is of great significance for water resources management and planning.

流量预报具有本质上的复杂性，动态性和动力学性，对水文学家提出了巨大的挑战。长短期记忆（LSTM）流量预测模型由于其强大的非线性建模能力而在近几年受到了广泛的关注。但是LSTM方法很少解决概率流量预测。在这项研究中，提出了结合高斯过程（GP）的概率长短期记忆网络来处理概率每日流量预测。此外，考虑到每日流量时间序列中存在随时间变化的平均值和方差，因此采用了异方差高斯过程回归来产生变化的预测间隔。所提出的方法封装了LSTM递归网络的电感偏差，并保留了高斯过程的非参数，概率性质。通过预测从长江上游及其支流收集的每日水流时间序列来研究所提出模型的性能。还开发了人工神经元网络，广义线性模型，异方差GP和常规LSTM模型进行比较。结果表明，该模型的性能令人满意。它提高了预测的准确性，并提供了一个自适应的预测间隔，这对水资源管理和规划具有重要意义。通过预测从长江上游及其支流收集的每日水流时间序列来研究所提出模型的性能。还开发了人工神经元网络，广义线性模型，异方差GP和常规LSTM模型进行比较。结果表明，该模型的性能令人满意。它提高了预测精度，并提供了一个自适应的预测间隔，这对水资源管理和规划具有重要意义。通过预测从长江上游及其支流收集的每日水流时间序列来研究所提出模型的性能。还开发了人工神经元网络，广义线性模型，异方差GP和常规LSTM模型进行比较。结果表明，该模型的性能令人满意。它提高了预测的准确性，并提供了一个自适应的预测间隔，这对水资源管理和规划具有重要意义。