Jul 2026· International Journal of Applied Mathematics, Computational Science and Systems Engineering· pp. 80· 0 citations· 4 references
TL;DR
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.
Abstract
Chaotic systems exhibit sensitivity to initial conditions and external parameters, posing challenges for long-term prediction. This study investigates the capability of deep neural networks (DNNs) to infer the time-evolution of the Rössler system a canonical chaotic oscillator by leveraging initial conditions (x0,y0,z0) and forcing parameters (a,b,c) as input variables. A 3D convolutional neural network (3D-CNN) architecture is designed to map these inputs to future states of the system. Results demonstrate that the DNN achieves high accuracy in short-term predictions (<50 time units) but faces exponential error growth beyond this horizon due to chaos. Notably, parameter variations (a,b,c) induce systematic shifts in attractor topology, while initial conditions amplify prediction uncertainty. The study highlights DNNs as viable tools for short-term chaotic forecasting but underscores the need for hybrid approaches to address long-term instability.
Deep learning surrogates for forecasting chaotic dynamical systems suffer from catastrophic error accumulation over long-term autoregressive rollouts. This behavior is partly tied to the underlying systems: chaotic spatiotemporal systems, such as the Kuramoto-Sivashinsky (KS) equation, visit phase space unevenly - dominated by recurrent, low-dimensional quiescent states (e.g., near-laminar flows) and punctuated by rare, dynamically complex topological transitions (e.g., wave-merging events). Under a sample-wise uniform objective, standard neural surrogates allocate their finite capacity to the statistically numerous quiescent states, under-representing the transient regimes that trigger disproportionate, localized errors. Existing imbalanced-regression methods reweight samples by target-space density. However, statistical target-space rarity need not coincide with the intrinsic dynamical rarity - the recurrence geometry of the attractor that is the source of the imbalance. To address this, we introduce Dynamics-Aware Weighting (DAW), a data-centric objective reweighting framework. Using the local dimension $d$ from dynamical systems theory as an a priori measure of a state's active degrees of freedom, DAW reshapes the loss landscape to allocate representational capacity toward the sparse, high-$d$ regimes where forecast errors are systematically large. On the chaotic KS equation, DAW consistently outperforms uniform training, purely statistical density weighting, and its randomly permuted ablation, reducing long-term autoregressive error relative to all baselines. Event-level analysis shows that DAW achieves this by suppressing the localized error amplifications incurred during sharp jumps in $d$, which accompany complex physical processes such as wave-merging in the KS system.
We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.
The world consists of many dynamical and chaotic systems whose underlying functions can be highly nonlinear. Traditional engineering and scientific methods emphasize simplifying problems, sometimes to the point that valuable information may be lost. Instead, understanding of these systems may require analysis of the underlying dynamics. In the past, the technique of discovering and analyzing strange attractors has yielded some success. However, much work is still needed. Machine learning (ML)-based models such as deep neural networks which are based on nonlinear functions may provide a great set of techniques to help in the discovery of structure in dynamical systems. In this paper, we propose an ML-based approach to search for structure in dynamical systems using Neural Input Optimization (NIO). We discuss our methodology and present promising results. Quantitative results help to validate our proposed NIO methodology for structure discovery.
R. A. Calix, Ajaykumar Rejith, Tae-Hoon Kim· Dynamics· 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.
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.
Accurate prediction of nonlinear dynamical systems becomes particularly challenging when the evolution of the dynamics depends on hidden, time-varying factors that are not directly observable. Although reservoir computing (RC) provides an efficient framework for modeling complex dynamics, standard approaches based on a single trained readout often experience reduced accuracy in such non-autonomous settings. We propose a multi-regime RC framework in which multiple readouts are trained under different dynamical conditions and combined through a short observation window to form a trajectory-dependent linear readout. This enables both regime identification and adaptation to unseen or intermediate dynamics. The method is evaluated on a Duffing oscillator with a time-varying forcing input and a Rössler system driven by chaotic forcing from a Chen system. The results show improved prediction accuracy compared to both regime-specific and single global models trained on data aggregated from multiple regimes.
S. Hadipour Lakmesari, H. Kantz, Francesco Sorrentino· Chaos· 0 citations