Table 1 Summary of previous studies and methods explained in sect. 2, alongside the dataset used and approach explanation.

Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication⁸

Mackey-glass

Original introduction of ESNs using large, randomly connected RNNs with fixed internal weights, where only output weights are trained using linear regression

Chaotic time series prediction using echo state network based on selective opposition grey wolf optimizer¹¹

Mackey-glass

Optimization of critical parameters including reservoir size, spectral radius, and weight matrix density using Selective Opposition Grey Wolf Optimizer (SOGWO)

Collective behavior of a small-world recurrent neural system with scale-free distribution¹²

Laser times series

Scale-free Highly-clustered Echo State Network (SHESN) replacing random reservoir connections with structured design incorporating small-world and scale-free characteristics

SCESN, SPESN, SWESN: Three recurrent neural echo state networks with clustered reservoirs for prediction of nonlinear and chaotic time series¹³

Laser times series

Three architectures (SCESN, SPESN, SWESN) clustering reservoir neurons using K-Means, PAM, and Ward algorithms, with mean neurons serving as backbone units

Design of a reservoir for cloud-enabled echo state network with high clustering coefficient¹⁴

Laser times series

High Clustered Echo State Network (HCESN) using evolutionary optimization algorithms (PSO, GA, DE) to cluster reservoir neurons into backbone and local neurons

Parameterizing echo state networks for multi-step time series prediction¹⁵

Mackey-glass

Systematic optimization of hyperparameters including reservoir size, density, spectral radius, leakage rate, regularization coefficient, initialization length, and training length

Laplacian Echo State Network for Multivariate Time Series Prediction¹⁶

Runoff and sunspots, temperature and rainfall

Laplacian Echo State Network (LAESN) using Laplacian eigenmaps to transform high-dimensional reservoir matrix into low-dimensional representation, solving ill-posed problems and reducing overfitting

Design of polynomial echo state networks for time series prediction¹⁷

Lorenz system, Nonlinear system identification, Sunspot series, NH4-N prediction

Polynomial Echo State Networks (PESN) extending ESNs with polynomial functions in output weights, using singular value decomposition for reservoir construction, and proposing both primal (P-PESN) and simplified (S-PESN) architectures

Impact of time‑history terms on reservoir dynamics and prediction accuracy in echo state networks¹⁸

Lorenz, Rössler time series

Study on impact of time-history terms (leaky integrator and chaotic neurons) in ESNs, showing they enhance delay capacity while maintaining diversity and stability of reservoir dynamics, leading to improved prediction accuracy