An official website of the United States government
Chaotic dynamics are ubiquitous in many real-world systems, ranging from biological and industrial processes to climate dynamics and the spread of viruses. These systems are characterized by high sensitivity to initial conditions, making it challenging to predict their future behavior confidently. In this study, we propose a novel deep-learning framework that addresses this challenge by directly exploiting the long-term compounding of local prediction errors during model training, aiming to extend the time horizon for reliable predictions of chaotic systems. Our approach observes the future trajectories of initial errors at a time horizon, modeling the evolution of the loss to that point through the use of two major components: 1) a recurrent architecture (Error Trajectory Tracing) designed to trace the trajectories of predictive errors through phase space, and 2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on three classic chaotic systems and six real-world time series prediction tasks with chaotic characteristics. The results show that our approach outperforms the state-of-the-art methods.
more »
« less
Widening the Time Horizon: Predicting the Long-Term Behavior of Chaotic Systems
The understanding of chaotic systems is challenging not only for theoretical research but also for many important applications. Chaotic behavior is found in many nonlinear dynamical systems, such as those found in climate dynamics, weather, the stock market, and the space-time dynamics of virus spread. A reliable solution for these systems must handle their complex space-time dynamics and sensitive dependence on initial conditions. We develop a deep learning framework to push the time horizon at which reliable predictions can be made further into the future by better evaluating the consequences of local errors when modeling nonlinear systems. Our approach observes the future trajectories of initial errors at a time horizon to model the evolution of the loss to that point with two major components: 1) a recurrent architecture, Error Trajectory Tracing, that is designed to trace the trajectories of predictive errors through phase space, and 2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on classic chaotic systems and real-world time series prediction tasks with chaotic characteristics, and show that our approach outperforms the current state-of-the-art methods.
more »
« less
- Award ID(s):
- 2008202
- PAR ID:
- 10464680
- Date Published:
- Journal Name:
- IEEE ICDM
- Page Range / eLocation ID:
- 833 to 842
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Chaos is a deterministic phenomenon that emerges under certain conditions in a nonlinear dynamic system when the trajectories of the state variables become periodic and highly sensitive to the initial conditions. Chaotic systems are flexible, and it has been shown that communication is possible using parametric feedback control. Chaos synchronization is the basis of using chaos in communication. Chaos synchronization refers to the characteristic that the trajectories of two identical chaotic systems, each with its own unique initial conditions, converge over time. In this paper, data extraction is performed on different chaotic equations implemented as circuits. Lorenz is the base system implemented in this paper, followed by Modified Lorenz, Chua’s, Lu¨’s, and Ro¨ssler systems. Additionally, more recent systems (e.g., SprottD Attractor) are included in the data extraction process. The robust system implementations provide an alternative to software chaos and architectures, and will further reduce the required power and area. These chaotic systems serve as alternatives for quantum era computing, which will cause synchronous and asynchronous techniques to fail. The data extracted organize different modes of chaos implementation based on the ease of their fabrication in integrated circuits. Performance metrics including power consumption, area, design load, noise, and robustness to process and temperature variant are extracted for each system to demonstrate a figure of merit. The figure of merit showcases chaos equations fitting to be implemented as a transmitter/receiver with a mode of chaotic ciphering in communication.more » « less
-
Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about incorrect physics in the presence of random noise and cannot easily handle the situation with incomplete data. In this paper, a new iterative learning algorithm for complex turbulent systems with partial observations is developed that alternates between identifying model structures, recovering unobserved variables, and estimating parameters. First, a causality-based learning approach is utilized for the sparse identification of model structures, which takes into account certain physics knowledge that is pre-learned from data. It has unique advantages in coping with indirect coupling between features and is robust to stochastic noise. A practical algorithm is designed to facilitate causal inference for high-dimensional systems. Next, a systematic nonlinear stochastic parameterization is built to characterize the time evolution of the unobserved variables. Closed analytic formula via efficient nonlinear data assimilation is exploited to sample the trajectories of the unobserved variables, which are then treated as synthetic observations to advance a rapid parameter estimation. Furthermore, the localization of the state variable dependence and the physics constraints are incorporated into the learning procedure. This mitigates the curse of dimensionality and prevents the finite time blow-up issue. Numerical experiments show that the new algorithm identifies the model structure and provides suitable stochastic parameterizations for many complex nonlinear systems with chaotic dynamics, spatiotemporal multiscale structures, intermittency, and extreme events.more » « less
-
Lacarbonara, Walter (Ed.)This work proposes a computational approach that has its roots in the early ideas of local Lyapunov exponents, yet, it offers new perspectives toward analyzing these problems. The method of interest, namely abstract dynamics, is an indirect quantitative measure of the variations of the governing vector fields based on the principles of linear systems. The examples in this work, ranging from simple limit cycles to chaotic attractors, are indicative of the new interpretation that this new perspective can offer. The presented results can be exploited in the structure of algorithms (most prominently machine learning algorithms) that are designed to estimate the complex behavior of nonlinear systems, even chaotic attractors, within their horizon of predictability.more » « less
-
Submitted paper. arxiv abstract: The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICDM54844.2022.00094
