Data-Driven Nonlinear System Identification Using Time Delayed Maps with Application to Multi-Attractor Systems

Alapfogalmak

This paper introduces a novel data-driven method called Nonlinear Delayed Maps (NLDM) for identifying nonlinear dynamical systems, particularly those with multiple attractors, by leveraging time-delayed states and nonlinear feature mappings to improve prediction accuracy across different regions of the phase space.

Kivonat

• Bibliographic Information: Iliopoulosa, A. P., Lunasinb, E., Michopoulosa, J. G., Rodrigueza, S. N., & Wigginsb, S.,c. (2024). Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors. Preprint submitted to Elsevier. arXiv:2411.10910v1 [math.DS]

• Research Objective: This paper presents a novel data-driven method for dynamical system identification, focusing on addressing the limitations of linear DMD methods in capturing nonlinear dynamics, particularly in systems with multiple attractors.

• Methodology: The proposed NLDM method combines aspects of EDMD and HODMD by incorporating time-delayed states and nonlinear feature mappings. The algorithm learns a linear operator that maps time-delayed and nonlinearly transformed state variables to future states. The method's performance is evaluated on a range of benchmark systems with varying complexity, including systems with single and multiple attractors.

• Key Findings: The NLDM algorithm demonstrates accurate prediction capabilities for a variety of nonlinear systems, including damped linear and nonlinear oscillators, a two-attractor system, and a damped double-well oscillator. The study highlights the importance of incorporating phase space geometry knowledge, particularly when dealing with multiple attractors. The authors demonstrate that training the algorithm with data from different basins of attraction significantly improves its ability to generalize and make accurate predictions across the entire phase space.

• Main Conclusions: The NLDM method offers a computationally efficient and robust approach for data-driven identification of nonlinear dynamical systems, even in the presence of noise. The incorporation of time-delayed states and nonlinear feature mappings, along with phase space-informed sampling, enables the algorithm to capture complex dynamics and generalize well across different regions of attraction.

• Significance: This research contributes to the field of system identification by providing a novel method that addresses limitations of existing linear approaches in capturing nonlinear dynamics. The NLDM algorithm's ability to handle multiple attractors and its robustness to noise make it a valuable tool for analyzing complex systems across various scientific and engineering domains.

• Limitations and Future Research: While the NLDM method shows promise, the paper acknowledges the challenge of identifying basin boundaries in practical applications. Future research could explore methods for automatically detecting and incorporating basin boundary information during the training process. Additionally, investigating the algorithm's performance on higher-dimensional and chaotic systems would further validate its capabilities and potential applications.

Statisztikák

The NLDM model achieved an RRMSE of 3.41 × 10−3 averaged over the training data set for the damped, linear harmonic oscillator.
On the independent test set for the damped, linear harmonic oscillator, the NLDM model produced an RRMSE of 3.26×10−3.
For the damped, double well oscillator, the prediction errors for the test trajectories are of similar magnitude, with an average RRMSE error of 1.5 × 10−9 and 5.7 × 10−7.

Idézetek

Főbb Kivonatok

Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors

by Athanasios P... : arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.10910.pdf

Mélyebb kérdések

How might the NLDM method be extended or adapted to handle systems with time-varying dynamics or external inputs?

The current NLDM method excels at modeling autonomous systems with time-invariant dynamics. However, many real-world systems are non-autonomous, meaning their behavior changes over time, often due to external inputs or time-dependent parameters. Here's how NLDM could be adapted for such cases:

Time-dependent NLDM (TD-NLDM): Instead of a single operator Λ, we could introduce time-dependence, making it Λ(t). This could be achieved by:

Parameterizing Λ: Represent Λ as a function of time with a suitable basis (e.g., Fourier series for periodic variations, polynomials for smooth trends). The learning process would then involve optimizing these parameters.
Time-windowing: Divide the data into smaller time windows where the dynamics are approximately stationary. Learn a separate Λ for each window, essentially creating a piecewise time-varying model.

Incorporating External Inputs: To handle systems of the form ẋ = f(x, u(t)), where u(t) represents external inputs:

Augment the Feature Vector: Include time-delayed versions of the input u(t) in the feature vector Υ. This allows the model to learn the mapping from both past states and inputs to future states.
Separate Input Operator: Introduce an additional operator specifically for the input, leading to a model like: xk = ΛxΥx + ΛuΥu, where Υx contains state features and Υu contains input features.

Recurrent Neural Networks (RNNs): For highly complex time-varying dynamics, RNNs offer a powerful alternative. The time-delayed states and nonlinear features in NLDM naturally lend themselves to an RNN architecture. The RNN could be trained on the same data used for NLDM, potentially capturing even more intricate temporal dependencies.

These extensions would require careful consideration of increased computational complexity and potential overfitting, especially with limited data. Regularization techniques and robust optimization methods would be crucial.

Could the reliance on phase space knowledge for systems with multiple attractors limit the applicability of NLDM in cases where such information is difficult or impossible to obtain a priori?

Yes, the reliance on phase space knowledge, particularly the basins of attraction, for systems with multiple attractors can indeed pose a limitation to the applicability of NLDM in cases where such information is not readily available.
Here's why:

Basin Boundaries are Often Unknown: In many real-world systems, the exact nature and location of basin boundaries are unknown a priori. Determining these boundaries analytically can be challenging or even impossible for high-dimensional, nonlinear systems.
Sampling Challenges: Without knowledge of the basins, it becomes difficult to ensure that the training data adequately samples all relevant regions of the phase space. This can lead to a model that is accurate within the sampled basins but fails to generalize to other unexplored regions.
Sensitivity to Initial Conditions: Systems with multiple attractors often exhibit sensitivity to initial conditions, especially near basin boundaries. Small perturbations in the initial state can lead to drastically different long-term behavior, making it crucial to have training data that captures this sensitivity.
Possible Mitigations:

Adaptive Sampling: Implement adaptive sampling techniques that iteratively explore the phase space and refine the training data based on the model's performance. This could involve starting with a coarse sampling and then focusing on regions where the model exhibits high prediction errors.
Ensemble Methods: Train an ensemble of NLDM models, each with a different initialization or training data subset. This can help to capture a wider range of dynamics and improve generalization.
Combining with Other Techniques: Integrate NLDM with other methods that do not rely on a priori phase space knowledge. For example, using unsupervised learning techniques like clustering to identify potential basins of attraction from the data itself.
While these mitigations can help to address the limitations, it's important to acknowledge that the lack of phase space knowledge can significantly complicate the modeling process and may require more data and computational resources to achieve satisfactory results.

How does the concept of attractors in dynamical systems relate to the idea of "attractor states" in other complex systems, such as social networks or economic models, and could methods like NLDM offer insights into these areas?

The concept of "attractors" in dynamical systems has intriguing parallels with "attractor states" in other complex systems like social networks, economic models, and even biological systems. While the mathematical formalism might differ, the underlying idea of stable, emergent patterns of behavior remains central.
Here's a breakdown:

Dynamical Systems: Attractors represent long-term tendencies of the system. A point in a basin of attraction, regardless of its initial state, will tend to evolve towards its corresponding attractor over time.
Complex Systems: "Attractor states" refer to stable, recurring patterns or configurations that a system tends to gravitate towards. For instance:

Social Networks: Stable patterns of opinion polarization, formation of echo chambers, or emergence of influential hubs.
Economic Models: Equilibrium states like stable prices, steady growth, or, conversely, persistent poverty traps.
Biological Systems: Stable cell types, recurring patterns in neural activity, or balanced states in ecosystems.
Could NLDM offer insights?
Potentially, yes. While NLDM is rooted in the mathematics of dynamical systems, its core principles of identifying underlying dynamics from time-series data could be adapted to study complex systems:

Identifying "Attractor States": By analyzing time-series data from social networks (e.g., sentiment scores, network connections) or economic indicators, NLDM-like approaches could help identify recurring patterns that resemble "attractor states."

Predicting Transitions: Understanding the dynamics around these "attractor states" might allow us to predict potential transitions or shifts. For example, forecasting a shift in public opinion or an economic downturn.

Policy Interventions: In a policy context, identifying these "attractor states" and their dynamics could help design interventions to steer a system towards a more desirable state.

Challenges and Considerations:

Data Complexity: Complex systems data is often noisy, high-dimensional, and fraught with nonlinear interactions, posing significant challenges for modeling.
Causality vs. Correlation: NLDM, like many data-driven methods, excels at identifying correlations in time-series data. Inferring causality in complex systems requires careful interpretation and domain expertise.
Ethical Considerations: Applying such models to social or economic systems raises ethical questions about manipulation, bias, and unintended consequences.
In conclusion, while there's promise in adapting methods like NLDM to study complex systems, it requires careful consideration of the unique challenges and ethical implications. Interdisciplinary collaborations between mathematicians, data scientists, and domain experts are crucial for responsible and insightful applications.