Aug 2026· Mathematics· Vol 14, pp. 2748· 0 citations· 187 references
TL;DR
This work uses error-free computation of two isomorphic chaotic systems, namely the Logistic map and the Tent map, to investigate the ability of Echo State Networks (ESNs) to learn and predict chaos, suggesting that ESNs exhibit significantly different predictive performance on the two isomorphic dynamical systems.
Abstract
We address the question of whether machine learning can improve the predictability of chaos. Due to the presence of chaos, chaotic time series are “contaminated” beyond the horizon of predictability as numerical errors accumulate. Therefore, evaluations should be based on error-free computation of chaos because learning the “contaminated” time series may not even be related to learning the chaotic dynamics. We use error-free computation of two isomorphic chaotic systems, namely the Logistic map and the Tent map, to investigate the ability of Echo State Networks (ESNs) to learn and predict chaos. ESNs exhibit significantly different predictive performance on the two isomorphic dynamical systems, suggesting that learning relies more on the arithmetic representation of the dynamical data than on the complexity or entropy production of the underlying dynamics. ESNs can achieve long horizons of predictability, but they do not improve the predictability of the corresponding dynamical models. Moreover, even when optimal predictive performance is achieved, this provides no guidance for other predictions within the same or its isomorphic dynamical system. The optimal hyperparameters depend strongly on the input time series and cannot be transferred, while increasing training length does not necessarily improve predictive performance.
Using ordinary least squares regression on high-degree polynomial features with 512-bit arithmetic, a system-agnostic method is introduced that matches the accuracy of standard 64-bit numerical ODE solvers using the systems’ governing equations, suggesting that forecasting low-dimensional chaotic systems from noise-free data is effectively a solved problem.
It is demonstrated that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.
Multi-step training of sparse, interpretable models of dynamical systems directly from time-series data yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents.
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.
Evrard Gabin Noutsa Tedjeuzen, Dagobert Wenkack Liedji, K. Lüdge et al.· AIP Advances· 0 citations
Machine learning of Hamiltonian dynamics has driven growing interest in Hamiltonian neural networks (HNNs), which encode Hamilton's equations of motion into the learning architecture. Despite this progress, it remains unknown whether such networks can predict dynamical regimes absent from their training data, in particular the broad chaotic sea that emerges beyond the observed parameter interval. We address this question using a parameter-aware random-feature Hamiltonian neural network (RF-HNN). Trained using data from only a small number of control-parameter values at which invariant tori dominate, the RF-HNN predicts autonomous long-time dynamics at unseen parameter values where mixed phase space develops and chaotic regions expand, with no data from that regime used in training or model selection. The method is demonstrated across four two-degree-of-freedom Hamiltonian families, including the H\'enon-Heiles system. Using Poincar\'e-section geometry and finite-time Lyapunov exponents, we show that the RF-HNN reproduces the breakup of regular structures and the emergence and growth of chaotic regions, whereas conventionally trained HNNs with the same Hamiltonian structure remain too regular. These results show that what decides parameter extrapolation is not Hamiltonian structure alone but how the fitted Hamiltonian continues in the control parameter. To our knowledge, this is the first demonstration that a learned Hamiltonian can qualitatively extrapolate from predominantly regular dynamics into a broad chaotic sea absent from training.
The edge-of-chaos heuristic has long served as a guiding principle for designing reservoir computers, yet its relevance to machine performance remains elusive. Here, taking the spectral radius of the reservoir network as the control parameter, we show that the radius yielding the best forecasting performance does not coincide with the Lyapunov edge of the isolated, teacher-forced, or closed-loop generative reservoir. By analyzing the collective dynamics of the teacher-forced reservoir, we find that the target dynamics are represented mainly by stable Lyapunov modes whose finite-time stability is strongly modulated by the input. This finding motivates a stability-expressivity transfer index, which balances the stability of these modes against their expressivity in representing the target. Across chaotic and quasiperiodic targets, and for both asymmetric and symmetric reservoirs, this index accurately identifies the optimal spectral radius for autonomous forecasting.
