Theoretical and experimental use of normalized mean square error (NMSE) from delay-based reservoir computer for dynamical characterization of systems

Abstract

The complexity of nonlinear dynamical systems can be analyzed using indicators such as Lyapunov exponents or bifurcation diagrams, whose estimation remains challenging and data-intensive in experimental contexts with constraints. At the same time, reservoir computing (RC) has emerged as a powerful tool for chaotic time series prediction, and the associated metric, the normalized mean square error (NMSE), is solely used to evaluate predictive performance. In this work, we demonstrate that the NMSE from an RC-based prediction model can be used as an indicator for system dynamical characterization. We propose a unified, numerical, and experimental approach based on a single-node time-delay reservoir computing architecture implemented around a Mackey-Glass oscillator and a low-cost hardware platform using an Arduino Due board. The approach is evaluated on discrete and continuous dynamical systems. The variations of NMSE as a function of control parameters are systematically compared with those of Lyapunov exponents and bifurcation diagrams. The results show that different transition regimes (periodic, chaotic, and hyperchaotic) can be reproducible in the variation of the NMSE, both numerically and experimentally. Viewed in this dynamical order, our results also reveal a sigmoidal relationship between the NMSE and the largest Lyapunov exponent. Therefore, the NMSE, beyond its classical role as a prediction metric, can be used as a data-driven indicator reflecting the complexity of a target system’s dynamics. This study thus provides an alternative approach for probing dynamical regimes from finite time series, paving the way for embedded neuromorphic tools dedicated to the experimental analysis of complex nonlinear systems.