# Deep Echo State Networks with Uncertainty Quantification for Spatio-Temporal Forecasting

arXiv: https://arxiv.org/abs/1806.10728 · HTML: https://mlsanity.com/paper/1806.10728/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2018.

## Built on by (most cited first, 16 of 16)

- [PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics](https://mlsanity.com/paper/2106.02993/) (2021)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Characterizing climate pathways using feature importance on echo state networks](https://mlsanity.com/paper/2310.08495/) (2023)
- [Echo State Networks for Spatio-Temporal Area-Level Data](https://mlsanity.com/paper/2410.10641/) (2024)
- [Evidential Deep Learning for Probabilistic Modelling of Extreme Storm Events](https://mlsanity.com/paper/2412.14048/) (2024)
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- [Recycling Scraps: Improving Private Learning by Leveraging Intermediate Checkpoints](https://mlsanity.com/paper/2210.01864/) (2022)
- [Probabilistic load forecasting with Reservoir Computing](https://mlsanity.com/paper/2308.12844/) (2023)
- [Survey on Deep Fuzzy Systems in regression applications: a view on interpretability](https://mlsanity.com/paper/2209.04230/) (2022)
- [Data-driven and machine-learning based prediction of wave propagation behavior in dam-break flood](https://mlsanity.com/paper/2209.08729/) (2022)
- [Variability of echo state network prediction horizon for partially observed dynamical systems](https://mlsanity.com/paper/2306.10797/) (2023)
- [Pedestrian Volume Prediction Using a Diffusion Convolutional Gated Recurrent Unit Model](https://mlsanity.com/paper/2411.03360/) (2024)
- [Modeling High-Resolution Spatio-Temporal Wind with Deep Echo State Networks and Stochastic Partial Differential Equations](https://mlsanity.com/paper/2412.07265/) (2024)
- [CESAR: A Convolutional Echo State AutoencodeR for High-Resolution Wind Forecasting](https://mlsanity.com/paper/2412.10578/) (2024)
- [Hierarchical Count Echo State Network Models with Application to Graduate Student Enrollments](https://mlsanity.com/paper/2501.14698/) (2025)
- [A Physics-Informed Spatiotemporal Deep Learning Framework for Turbulent Systems](https://mlsanity.com/paper/2505.10919/) (2025)

## Abstract (arXiv)

Long-lead forecasting for spatio-temporal systems can often entail complex nonlinear dynamics that are difficult to specify it 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 use so-called it reservoir computing to efficiently compute recurrent neural network (RNN) forecasts. Moreover, so-called "deep" 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 we often find in spatio-temporal environmental data). Here we introduce a deep ensemble ESN (D-EESN) model. We present two versions of this model for spatio-temporal processes that both 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.