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Area of Science:

  • Dynamical Systems Theory
  • Machine Learning
  • Time-Series Analysis

Background:

  • Predicting system dynamics from partial observations is a significant challenge in time-series analysis.
  • Takens' theorem provides a theoretical basis for time-delayed embeddings but learning these mappings for complex systems is difficult.
  • Existing recurrent neural network models often require high-dimensional internal states or numerous hyperparameters.

Purpose of the Study:

  • To utilize deep artificial neural networks (ANNs) for learning discrete and continuous time dynamics from partial observations of a system.
  • To develop a method for predicting time series using learned embeddings and reconstruction maps.
  • To demonstrate the effectiveness of ANNs in capturing complex dynamics from limited data.

Main Methods:

  • Employed deep artificial neural networks (ANNs) to learn discrete time maps and continuous time flows from partial state observations.
  • Developed a reconstruction map using full state training data.
  • Applied time-series analysis to determine embedding parameters for prediction.
  • Validated the approach using the Lorenz system (chaotic dynamics from scalar observation) and the Kuramoto-Sivashinsky equation (multivariate observations).

Main Results:

  • Deep ANNs successfully learned discrete and continuous time dynamics from partial observations.
  • The method enabled accurate time-series predictions using learned embeddings and reconstruction maps.
  • Demonstrated effective prediction of chaotic behavior in the Lorenz system from a single scalar observation.
  • Showcased the ability to reproduce dynamics for the Kuramoto-Sivashinsky equation with multivariate observations, highlighting the relationship between observation dimension and system complexity.

Conclusions:

  • Deep ANNs offer a powerful tool for learning and predicting the dynamics of complex systems from partial or scalar time-series data.
  • This approach provides an alternative to traditional methods, potentially reducing the need for high-dimensional internal states or extensive hyperparameter tuning.
  • The findings suggest ANNs can effectively reconstruct and predict system evolution on low-dimensional manifolds, even for chaotic and nonlinear systems.