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Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications. 

Abstract Long‐lead forecasting for spatio‐temporal systems can entail complex nonlinear dynamics that are difficult to specify a priori. Current statistical methodologies for modeling these processes are often highly parameterized and, thus, challenging to implement from a computational perspective. One potential parsimonious solution to this problem is a method from the dynamical systems and engineering literature referred to as an echo state network (ESN). ESN models usereservoir computingto efficiently compute recurrent neural network forecasts. Moreover, multilevel (deep) hierarchical models have recently been shown to be successful at predicting high‐dimensional complex nonlinear processes, particularly those with multiple spatial and temporal scales of variability (such as those we often find in spatio‐temporal environmental data). Here, we introduce a deep ensemble ESN (D‐EESN) model. Despite the incorporation of a deep structure, the presented model is computationally efficient. We present two versions of this model for spatio‐temporal processes that produce forecasts and associated measures of uncertainty. The first approach utilizes a bootstrap ensemble framework, and the second is developed within a hierarchical Bayesian framework (BD‐EESN). This more general hierarchical Bayesian framework naturally accommodates non‐Gaussian data types and multiple levels of uncertainties. The methodology is first applied to a data set simulated from a novel non‐Gaussian multiscale Lorenz‐96 dynamical system simulation model and, then, to a long‐lead United States (U.S.) soil moisture forecasting application. Across both applications, the proposed methodology improves upon existing methods in terms of both forecast accuracy and quantifying uncertainty.