Hybridizing multiscale and delayed reservoir computing for long-term accurate chaotic system forecasting

Prediction of chaotic dynamical systems remains a fundamental challenge, primarily due to their extreme sensitivity to initial conditions and the coexistence of multiscale temporal dynamics. Although traditional Reservoir Computing (RC) and its delayed-feedback extension (DRC) have demonstrated promising performance in chaotic system prediction, they often struggle to capture the complex multiscale features present in such systems. To address these limitations, a Hybrid MultiScale Delayed Reservoir Computing framework (HyMS-DRC) is proposed. This approach integrates standard RC and delayed-feedback RC in a parallel multiscale architecture, where state fusion enhances the representation of dynamics across different temporal scales and mitigates memory fading in long-horizon prediction. We systematically evaluate HyMS-DRC on three canonical chaotic systems-Double Scroll, Lorenz, and Rössler-and compare it against RC, DRC, MultiScale Reservoir Computing (MS-RC), and MultiScale Delayed-Feedback Reservoir Computing (MS-DRC). Experimental results demonstrate that HyMS-DRC consistently achieves superior forecasting performance, attaining the lowest normalized root-mean-square errors (NRMSEs) of 0.0089, 0.0137, and 0.0662, and the longest valid prediction times of approximately 27.85, 24.29, and 22.37 on the three systems, respectively. Furthermore, long-term statistical analyses confirm that HyMS-DRC robustly reconstructs the attractor geometry and accurately reproduces the power spectral distribution, indicating excellent generalization and modeling capability for complex nonlinear dynamics. These findings highlight that combining multiscale structures with delayed feedback significantly enhances temporal memory and feature representation in reservoirs, enabling accurate and robust long-term prediction of chaotic systems.