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Reconstruction, forecasting, and stability of chaotic dynamics from partial data.

Elise Özalp1, Georgios Margazoglou1, Luca Magri1,2

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Data-driven methods using long short-term memory (LSTM) networks can reconstruct chaotic system dynamics and stability from partial observations. These methods accurately forecast hidden variables and infer stability properties, outperforming traditional approaches.

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Area of Science:

  • Computational physics and applied mathematics.
  • Dynamical systems theory.
  • Machine learning for scientific discovery.

Background:

  • Traditional equation-based methods struggle with forecasting and stability computation in chaotic systems using partial data.
  • Reconstructing hidden dynamics and inferring stability from limited observations is a significant challenge.

Purpose of the Study:

  • To develop and evaluate data-driven methods for full-state reconstruction and stability analysis of chaotic systems from partial observations.
  • To infer hidden chaotic dynamics and forecast system evolution using machine learning.
  • To compute stability properties like Lyapunov exponents from incomplete data.

Main Methods:

  • Utilized long short-term memory (LSTM) networks, including low-to-high resolution LSTM (LH-LSTM) and physics-informed LSTM (PI-LSTM).
  • Trained LSTMs with partial state observations, incorporating system dynamics equations for PI-LSTM.
  • Derived Jacobian of LSTMs and analyzed chaotic systems like Kuramoto-Sivashinsky and Lorenz-96.

Main Results:

  • Proposed LSTM networks successfully forecast hidden variables in chaotic systems with time-accuracy and statistical fidelity.
  • Lyapunov exponents and covariant Lyapunov vectors, key stability indicators, were correctly inferred from partial observations.
  • Physics-informed LSTM (PI-LSTM) demonstrated superior performance in reconstructing hidden dynamics, especially with limited input dimensions and noisy data.

Conclusions:

  • Data-driven LSTM methods offer a viable alternative to traditional approaches for chaotic system analysis from partial data.
  • The developed PI-LSTM effectively reconstructs hidden dynamics and infers stability properties, advancing the field of chaotic system modeling.
  • This work provides new avenues for inferring hidden variables and computing stability in complex systems using incomplete observational data.