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.
Abstract
Low-dimensional chaotic systems such as the Lorenz-63 model are commonly used to benchmark system-agnostic methods for learning dynamics from data. This study shows that learning from noise-free observations in such systems can be achieved up to machine precision: 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. For the Lorenz-63 system, the method achieves valid prediction times of 36 Lyapunov times, and even up to 105 Lyapunov times with favorable precision configurations, dramatically outperforming prior work, which reaches 13 Lyapunov times at most. The results are further validated on Thomas’ Cyclically Symmetric Attractor, a non-polynomial chaotic system that is considerably more complex than the Lorenz-63 model, and similar results extend to higher dimensions using the spatiotemporally chaotic Lorenz-96 model. These findings suggest that forecasting low-dimensional chaotic systems from noise-free data is effectively a solved problem.
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
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.
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.
Alexandros K. Angelidis, Georgios C. Makris, E. Ioannidis et al.· Mathematics· 0 citations
Largest Lyapunov exponents (LLEs) quantify exponential sensitivity, but data-driven estimates are often obtained from finite-precision trajectories. We show that increasing the length of a single reduced-precision chaotic record can eventually degrade a forecast-error LLE estimate. Using the logistic map at r=4, an ESP32 single-precision trajectory is reproduced bit-for-bit by NumPy float32. Across 10,000 random float32 initial conditions, every trajectory reaches an exact recurrence before iteration 7612. For one long float32 record, the estimated LLE changes from 0.6853 at N=15,000 to 0.1827 at N=20,000 and approximately zero at N=30,000 as exact train-test histories saturate. At N=100,000, the long float32 record gives 0.0016, whereas independently restarted length-100 trajectories give 0.6917; matched float64 controls remain near ln(2)=0.6931. The collapse is reproduced for 28 representative initial conditions, and its onset is strongly correlated with the recurrence scale set by transient length and digital period (Pearson r=0.982). Thus, finite-state recurrence can turn additional samples into duplicate futures rather than new dynamical information, while independent restarts substantially delay this saturation.
This study investigates the capability of deep neural networks to infer the time-evolution of the Rössler system a canonical chaotic oscillator by leveraging initial conditions and forcing parameters as input variables and underscores the need for hybrid approaches to address long-term instability.
A. Fateh, Harrag Abdelmalek, F. Mohamed et al.· International Journal of App...· 0 citations
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.