We first study the generalization error of models that use a fixed feature representation (frozen intermediate layers) followed by a trainable readout layer. This setting encompasses a range of architectures, from deep random-feature models to echo-state networks (ESNs) with recurrent dynamics. Working in the high-dimensional regime, we apply Random Matrix Theory to derive a closed-form expression for the asymptotic generalization error. We then apply this analysis to recurrent representations and obtain concise formula that characterize their performance. Surprisingly, we show that a linear ESN is equivalent to ridge regression with an exponentially time-weighted (''memory'') input covariance, revealing a clear inductive bias toward recent inputs. Experiments match predictions: ESNs win in low-sample, short-memory regimes, while ridge prevails with more data or long-range dependencies. Our methodology provides a general framework for analyzing overparameterized models and offers insights into the behavior of deep learning networks.

In this work, a three-dimensional multicomponent reaction-diffusion model has been developed, combining excitable-system dynamics with diffusion processes and sharing conceptual features with the FitzHugh-Nagumo model. Designed to capture the spatiotemporal behavior of biological systems, particularly electrophysiological processes, the model was solved numerically to generate time-series data. These data were subsequently used to train and evaluate an Echo State Network (ESN), which successfully reproduced the system's dynamic behavior. The results demonstrate that simulating biological dynamics using data-driven, multifunctional ESN models is both feasible and effective.

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

This paper examines Echo State Network, a reservoir computer, performance using four different benchmark problems, then proposes heuristics or rules of thumb for configuring the architecture, as well as the selection of parameters and their values, which are applicable to problems within the same domain, to help serve to fill the experience gap needed by those entering this field of study. The influence of various parameter selections and their value adjustments, as well as architectural changes made to an Echo State Network, a powerful recurrent neural network configured as a reservoir computer, can be challenging to fully comprehend without experience in the field, and even some hyperparameter optimization algorithms may have difficulty adjusting parameter values without proper manual selections made first. Therefore, it is imperative to understand the effects of parameters and their value selection on Echo State Network architecture performance for a successful build. Thus, to address the requirement for an extensive background in Echo State Network architecture, as well as examine how Echo State Network performance is affected with respect to variations in architecture, design, and parameter selection and values, a series of benchmark tasks representing different problem domains, including time series prediction, pattern generation, chaotic system prediction, and time series classification, were modeled and experimented on to show the impact on the performance of Echo State Network.

Forecasting stock and cryptocurrency prices is challenging due to high volatility and non-stationarity, influenced by factors like economic changes and market sentiment. Previous research shows that Echo State Networks (ESNs) can effectively model short-term stock market movements, capturing nonlinear patterns in dynamic data. To the best of our knowledge, this work is among the first to explore ESNs for cryptocurrency forecasting, especially during extreme volatility. We also conduct chaos analysis through the Lyapunov exponent in chaotic periods and show that our approach outperforms existing machine learning methods by a significant margin. Our findings are consistent with the Lyapunov exponent analysis, showing that ESNs are robust during chaotic periods and excel under high chaos compared to Boosting and Naïve methods.

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.

With the deep penetration of Artificial Intelligence (AI) in the transportation sector, intelligent cockpits, autonomous driving, and intelligent road networks are developing at an unprecedented pace. However, the data ecosystems of these three key areas are increasingly fragmented and incompatible. Especially, existing testing methods rely on data stacking, fail to cover all edge cases, and lack flexibility. To address this issue, this paper introduces the concept of "Query as Test" (QaT). This concept shifts the focus from rigid, prescripted test cases to flexible, on-demand logical queries against a unified data representation. Specifically, we identify the need for a fundamental improvement in data storage and representation, leading to our proposal of "Extensible Scenarios Notations" (ESN). ESN is a novel declarative data framework based on Answer Set Programming (ASP), which uniformly represents heterogeneous multimodal data from the cockpit, vehicle, and road as a collection of logical facts and rules. This approach not only achieves deep semantic fusion of data, but also brings three core advantages: (1) supports complex and flexible semantic querying through logical reasoning; (2) provides natural interpretability for decision-making processes; (3) allows for on-demand data abstraction through logical rules, enabling fine-grained privacy protection. We further elaborate on the QaT paradigm, transforming the functional validation and safety compliance checks of autonomous driving systems into logical queries against the ESN database, significantly enhancing the expressiveness and formal rigor of the testing. Finally, we introduce the concept of "Validation-Driven Development" (VDD), which suggests to guide developments by logical validation rather than quantitative testing in the era of Large Language Models, in order to accelerating the iteration and development process.

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.