Model-free analysis of complex systems using delayed-feedback echo state network

Investigating the dynamical behavior of complex systems from data poses formidable challenges in the field of nonlinear science. This paper proposes a delayed-feedback echo state network (df-ESN) model specifically designed for modeling and analyzing complex systems from data. The df-ESN introduces the delayed feedback of the reservoir state to reflect the finite transmission speed of signals among reservoir units. The delayed feedback term links the current reservoir state to its value m steps ago, thereby significantly enhancing the memory capacity. By adopting the Lyapunov-Krasovskii stability method, we establish delay-dependent stability criteria to ensure the global echo state property (ESP). Furthermore, the definition of local echo state property (local ESP) is proposed to weaken the restrictive constraints imposed by the conventional ESP condition. The df-ESN also offers a strategy to determine the optimal parameters in the delayed feedback mechanism via a grid search technique, thus overcoming the challenges of determining optimal reservoir parameters in the ESN. This work provides a comprehensive theoretical understanding for the memory capability, local/global echo state property, and the impact of the noise on the model performance. Various nonlinear dynamical problems are conducted to verify the effectiveness of df-ESN, including chaotic time series prediction, high-dimensional spatiotemporal system analysis, bifurcation diagram reconstruction, and basin of attraction prediction. Moreover, df-ESN is used to study the model-free dynamical analysis of double soliton solutions of the nonlinear Schrödinger equation. The results underscore the capability of the proposed df-ESN as a model-free strategy for analyzing the complex systems, and they contribute to a deeper understanding of the underlying theoretical aspects of the model.