Abstract
Echo State Networks (ESNs) have emerged as powerful tools for time series prediction, yet their performance heavily depends on reservoir structure, which traditionally relies on random weights independent of input data characteristics. This paper presents a theoretical framework and optimization approach for ESN reservoir design, demonstrating that reservoir weights should be adapted based on input data properties, and that both topology and weights significantly influence prediction accuracy. Building on theoretical insights about input-dependent reservoir behavior, we propose two complementary methods: a supervised approach that directly optimizes reservoir weights through gradient descent, and a semi-supervised technique that combines small-world and scale-free network properties with hyperparameter optimization. Our extensive experiments across multiple datasets, including synthetic chaotic systems (Mackey-Glass and NARMA time series) and real-world climate data, demonstrate that the proposed methods consistently outperform traditional ESNs by achieving substantially lower prediction errors. Most notably, our analysis reveals that edge connectivity parameters play a crucial role, second only to reservoir size in determining network performance. These findings provide important practical guidelines for ESN design and open new directions for automated reservoir optimization based on input data characteristics.