Deep multi-modal learning for joint linear representation of nonlinear dynamical systems.

Area of Science:

• Complex Systems Science
• Dynamical Systems Theory
• Machine Learning

Background:

• Dynamical systems are prevalent in real-world applications, exhibiting complex linear, non-linear, or stochastic behaviors.
• Analyzing multi-modal observations (e.g., vital signs, EEG) is crucial for understanding system dynamics, but complexity and time-delays pose challenges.
• Existing methods struggle to directly characterize system dynamics due to the complexity of individual modalities and inter-modal interactions.

Purpose of the Study:

• To propose a novel deep auto-encoder framework leveraging Koopman operator theory.
• To derive joint linear dynamics for a target system within a shared intrinsic coordinate space.
• To reconstruct system states, restore missing observations, and predict future states using multi-modal data.

Main Methods:

• Developed a deep auto-encoder architecture integrated with Koopman operator theory.
• Trained the framework to learn intrinsic dynamics shared across multiple data modalities.
• Utilized the learned dynamics for state reconstruction, imputation, and prediction.

Main Results:

• Successfully reconstructed original system states by learning cross-modal information.
• Demonstrated the capability to restore missing observations within and across modalities.
• Enabled accurate prediction of future system states based on the derived intrinsic dynamics.

Conclusions:

• The proposed framework effectively captures the intrinsic dynamics of complex systems from multi-modal observations.
• This approach offers a powerful tool for system understanding, prediction, and data completion in various applications.
• The integration of deep learning and Koopman theory provides a unified method for analyzing complex dynamical systems.