We present a rigorous asymptotic analysis of Echo State Networks (ESNs) in a teacher student setting with a linear teacher with oracle weights. Leveraging random matrix theory, we derive closed form expressions for the asymptotic bias, variance, and mean-squared error (MSE) as functions of the input statistics, the oracle vector, and the ridge regularization parameter. The analysis reveals two key departures from classical ridge regression: (i) ESNs do not exhibit double descent, and (ii) ESNs attain lower MSE when both the number of training samples and the teacher memory length are limited. We further provide an explicit formula for the optimal regularization in the identity input covariance case, and propose an efficient numerical scheme to compute the optimum in the general case. Together, these results offer interpretable theory and practical guidelines for tuning ESNs, helping reconcile recent empirical observations with provable performance guarantees

Using data from mobile network utilization in Norway, we showcase the possibility of monitoring the state of communication and mobility networks with a non-invasive, low-cost method. This method transforms the network data into a model within the framework of reservoir computing and then measures the model's performance on proxy tasks. Experimentally, we show how the performance on these proxies relates to the state of the network. A key advantage of this approach is that it uses readily available data sets and leverages the reservoir computing framework for an inexpensive and largely agnostic method. Data from mobile network utilization is available in an anonymous, aggregated form with multiple snapshots per day. This data can be treated like a weighted network. Reservoir computing allows the use of weighted, but untrained networks as a machine learning tool. The network, initialized as a so-called echo state network (ESN), projects incoming signals into a higher dimensional space, on which a single trained layer operates. This consumes less energy than deep neural networks in which every weight of the network is trained. We use neuroscience inspired tasks and trained our ESN model to solve them. We then show how the performance depends on certain network configurations and also how it visibly decreases when perturbing the network. While this work serves as proof of concept, we believe it can be elevated to be used for near-real-time monitoring as well as the identification of possible weak spots of both mobile communication networks as well as transportation networks.

We propose an online learning framework for forecasting nonlinear spatio-temporal signals (fields). The method integrates (i) dimensionality reduction, here, a simple proper orthogonal decomposition (POD) projection; (ii) a generalized autoregressive model to forecast reduced dynamics, here, a reservoir computer; (iii) online adaptation to update the reservoir computer (the model), here, ensemble sequential data assimilation. We demonstrate the framework on a wake past a cylinder governed by the Navier-Stokes equations, exploring the assimilation of full flow fields (projected onto POD modes) and sparse sensors. Three scenarios are examined: a naïve physical state estimation; a two-fold estimation of physical and reservoir states; and a three-fold estimation that also adjusts the model parameters. The two-fold strategy significantly improves ensemble convergence and reduces reconstruction error compared to the naïve approach. The three-fold approach enables robust online training of partially-trained reservoir computers, overcoming limitations of a priori training. By unifying data-driven reduced order modelling with Bayesian data assimilation, this work opens new opportunities for scalable online model learning for nonlinear time series forecasting.

Machine learning (ML) is widely used to model chaotic systems. Among ML approaches, echo state networks (ESNs) have received considerable attention due to their simple construction and fast training. However, ESN performance is highly sensitive to hyperparameter choices and to its random initialization. In this work, we demonstrate that ESNs constructed using deterministic rules and simple topologies (MESNs) outperform standard ESNs in the task of chaotic attractor reconstruction. We use a dataset of more than 90 chaotic systems to benchmark 10 different minimal deterministic reservoir initializations. We find that MESNs obtain up to a 41% reduction in error compared to standard ESNs. Furthermore, we show that the MESNs are more robust, exhibiting less inter-run variation, and have the ability to reuse hyperparameters across different systems. Our results illustrate how structured simplicity in ESN design can outperform stochastic complexity in learning chaotic dynamics.

In this paper, we explore the predictive capabilities of echo state networks (ESNs) for the generalized Kuramoto-Sivashinsky (gKS) equation, an archetypal nonlinear PDE that exhibits spatiotemporal chaos. We introduce a novel methodology that integrates ESNs with transfer learning, aiming to enhance predictive performance across various parameter regimes of the gKS model. Our research focuses on predicting changes in long-term statistical patterns of the gKS model that result from varying the dispersion relation or the length of the spatial domain. We use transfer learning to adapt ESNs to different parameter settings and successfully capture changes in the underlying chaotic attractor.

What is a neural model with minimum architectural complexity that exhibits reasonable language learning capability? To explore such a simple but sufficient neural language model, we revisit a basic reservoir computing (RC) model, Echo State Network (ESN), a restricted class of simple Recurrent Neural Networks. Our experiments showed that ESN with a large hidden state is comparable or superior to Transformer in grammaticality judgment tasks when trained with about 100M words, suggesting that architectures as complex as that of Transformer may not always be necessary for syntactic learning.

--The unprecedented dissemination of edge devices is accompanied by a growing demand for neuromorphic chips that can process time-series data natively without cloud support. Echo state network (ESN) is a class of recurrent neural networks that can be used to identify unique patterns in time-series data and predict future events. It is known for minimal computing resource requirements and fast training, owing to the use of linear optimization solely at the readout stage. In this work, a custom-design neuromorphic chip based on ESN targeting edge devices is proposed. The proposed system supports various learning mechanisms, including structural plasticity and synaptic plasticity, locally on-chip. This provides the network with an additional degree of freedom to continuously learn, adapt, and alter its structure and sparsity level, ensuring high performance and continuous stability. We demonstrate the performance of the proposed system as well as its robustness to noise against real-world time-series datasets while considering various topologies of data movement. An average accuracy of 95.95% and 85.24% are achieved on human activity recognition and prosthetic finger control, respectively. HE last decade has seen significant advancement in neuromorphic computing with a major thrust centered around processing streaming data using recurrent neural networks (RNNs). Despite the fact RNNs demonstrate promising performance in numerous domains including speech recognition [1], computer vision [2], stock trading [3], and medical diagnosis [4], such networks suffer from slow convergence and intensive computations [5]. In order to bypass these challenges, Jaeger and Maass suggest leveraging the rich dynamics offered by the networks' recurrent connections and random parameters and limit the training to the network advanced layers, particularly the readout layer [7]-[9]. With that, the network training and its computation complexity are significantly simplified. There are three classes of RNN networks trained using this approach known as a liquid state machine (LSM) [7], delayed-feedback reservoir [10], [11], and echo state network (ESN) which is going to be the focus of this work. ESN is demonstrated in a variety of tasks, including pattern recognition, anomaly detection [12], spatial-temporal forecasting [13], and modeling dynamic motions in bio-mimic robots [14].

Poisson autoregressive count models have evolved into a time series staple for correlated count data. This paper proposes an alternative to Poisson autoregressions: count echo state networks. Echo state networks can be statistically analyzed in frequentist manners via optimizing penalized likelihoods, or in Bayesian manners via MCMC sampling. This paper develops Poisson echo state techniques for count data and applies them to a massive count data set containing the number of graduate students from 1,758 United States universities during the years 1972-2021 inclusive. Negative binomial models are also implemented to better handle overdispersion in the counts. Performance of the proposed models are compared via their forecasting performance as judged by several methods. In the end, a hierarchical negative binomial based echo state network is judged as the superior model.

An accurate and timely assessment of wind speed and energy output allows an efficient planning and management of this resource on the power grid. Wind energy, especially at high resolution, calls for the development of nonlinear statistical models able to capture complex dependencies in space and time. This work introduces a Convolutional Echo State AutoencodeR (CESAR), a spatio-temporal, neural network-based model which first extracts the spatial features with a deep convolutional autoencoder, and then models their dynamics with an echo state network. We also propose a two-step approach to also allow for computationally affordable inference, while also performing uncertainty quantification. We focus on a high-resolution simulation in Riyadh (Saudi Arabia), an area where wind farm planning is currently ongoing, and show how CESAR is able to provide improved forecasting of wind speed and power for proposed building sites by up to 17% against the best alternative methods.

The data-driven learning of solutions of partial differential equations can be based on a divide-and-conquer strategy. First, the high dimensional data is compressed to a latent space with an autoencoder; and, second, the temporal dynamics are inferred on the latent space with a form of recurrent neural network. In chaotic systems and turbulence, convolutional autoencoders and echo state networks (CAE-ESN) successfully forecast the dynamics, but little is known about whether the stability properties can also be inferred. We show that the CAE-ESN model infers the invariant stability properties and the geometry of the tangent space in the low-dimensional manifold (i.e. the latent space) through Lyapunov exponents and covariant Lyapunov vectors. This work opens up new opportunities for inferring the stability of high-dimensional chaotic systems in latent spaces.