Reservoir Computing via Multi-Scale Random Fourier Features for Forecasting Fast-Slow Dynamical Systems

TL;DR

This paper introduces a reservoir computing method using multi-scale random Fourier features to effectively forecast complex nonlinear systems with fast-slow dynamics, outperforming single-scale approaches across various models.

Contribution

The paper proposes a novel multi-scale RFF reservoir computing framework that captures multi-scale temporal dependencies in complex dynamical systems, improving forecasting accuracy.

Findings

Multi-scale RFF reservoir outperforms single-scale in forecasting accuracy.

The framework effectively models fast-slow interactions in diverse systems.

Multi-scale approach yields more robust long-term predictions.

Abstract

Forecasting nonlinear time series with multi-scale temporal structures remains a central challenge in complex systems modeling. We present a novel reservoir computing framework that combines delay embedding with random Fourier feature (RFF) mappings to capture such dynamics. Two formulations are investigated: a single-scale RFF reservoir, which employs a fixed kernel bandwidth, and a multi-scale RFF reservoir, which integrates multiple bandwidths to represent both fast and slow temporal dependencies. The framework is applied to a diverse set of canonical systems: neuronal models such as the Rulkov map, Izhikevich model, Hindmarsh-Rose model, and Morris-Lecar model, which exhibit spiking, bursting, and chaotic behaviors arising from fast-slow interactions; and ecological models including the predator-prey dynamics and Ricker map with seasonal forcing, which display multi-scale oscillations and intermittency. Across all cases, the multi-scale RFF reservoir consistently outperforms its single-scale counterpart, achieving lower normalized root mean square error (NRMSE) and more robust long-horizon predictions. These results highlight the effectiveness of explicitly incorporating multi-scale feature mappings into reservoir computing architectures for modeling complex dynamical systems with intrinsic fast-slow interactions.