Multiscale deep reservoir computing for predicting chaotic dynamical systems

We propose a Multiscale Deep Reservoir Computing (MSDRC) framework for predicting complex nonlinear dynamical systems. The framework incorporates a k-hop information propagation mechanism into deep reservoir architectures, aligning multi-hop state interactions with the hierarchical organization of sub-reservoirs to represent system dynamics across multiple temporal scales. Based on this principle, four MSDRC-based reservoir computer variants-deepMSFESN, deepMSBESN, groupedMSESN, and deepMSESN-are developed to achieve hierarchical multiscale feature fusion. Experiments on the Hindmarsh-Rose and Lorenz-63 systems demonstrate that MSDRC achieves higher predictive accuracy, robustness, and generalization under different initial conditions than standard and deep reservoir computing models. Parameter analyses further indicate that sparse reservoirs can still generate rich dynamics, while increasing reservoir size yields diminishing but consistent improvements in prediction performance. DeepESNs and MSDRC also outperform standard ESN and simple cycle reservoir models under varying slow timescale parameters, with MSDRC maintaining lower prediction errors and modest improvements by more effectively capturing the coupling between fast and slow dynamics. The sampling interval plays a critical role: smaller intervals improve predictive accuracy but require more steps to reach the same prediction time, thereby amplifying error accumulation in closed-loop operation. In contrast, larger intervals reduce temporal resolution and fail to capture system dynamics, revealing an inherent trade-off. Overall, MSDRC provides an effective and structurally interpretable multiscale framework for chaotic time series prediction and offers new insights into multiscale information fusion in reservoir computing.