This paper introduces a novel method for time series forecasting by leveraging Quantum-inspired Non-dominated Sorting Genetic Algorithm II (QNSGA-II) to optimize Echo State Networks (ESN), complemented by the evaluation of reservoir dynamics through the Separation Ratio Graph (SRG). The integration of QNSGA-II enables the simultaneous optimization of multiple ESN hyperparameters, aiming to reduce forecast error while enhancing the diversity and performance of the reservoir. SRG is employed as a metric to assess the quality of the reservoir's internal states, allowing for the identification of configurations that improve the model's ability to capture the complex dynamics inherent in time series data.

The proposed approach is validated using the Mackey-Glass time series dataset, a benchmark known for its nonlinear dynamics. Results show that the QNSGA-II optimized ESN with SRG evaluation significantly outperforms traditional ESN models, yielding a lower Mean Squared Error (MSE) in predictive performance. Additionally, the use of SRG offers deeper insights into reservoir behavior, facilitating more informed decision-making in the selection of optimal configurations.

The combination of QNSGA-II and SRG not only enhances the robustness of the ESN but also provides a comprehensive framework for improving the accuracy and reliability of time series forecasting. This method represents a step forward in leveraging quantum-inspired optimization techniques for neural networks, demonstrating the potential of hybrid approaches in addressing the challenges of nonlinear and chaotic time series prediction.