Stability analysis of chaotic systems in latent spaces

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning

Background:

• Partial differential equations (PDEs) model complex systems, often exhibiting chaotic solutions.
• Data-driven methods offer a novel approach to solving PDEs by inferring dynamics in a compressed latent space.
• Traditional methods can be computationally intensive for chaotic systems.

Purpose of the Study:

• To demonstrate that a latent-space approach can solve chaotic PDEs and predict system stability.
• To apply the convolutional autoencoder echo state network (CAE-ESN) to chaotic systems.
• To validate the CAE-ESN's ability to infer Lyapunov exponents and covariant Lyapunov vectors (CLVs).

Main Methods:

• Utilizing a convolutional autoencoder echo state network (CAE-ESN) for data compression and temporal dynamics inference.
• Applying the CAE-ESN to the chaotic Kuramoto-Sivashinsky equation across various chaotic regimes.
• Extending the CAE-ESN to analyze turbulent flow dynamics and comparing results with Jacobian-free methods.

Main Results:

• The CAE-ESN successfully identified a low-dimensional latent-space representation of observational data.
• Accurate inference of Lyapunov exponents and CLVs within the low-dimensional manifold for different attractors.
• The model effectively preserved the geometric structure of the chaotic system's attractor.

Conclusions:

• The latent-space approach based on CAE-ESN serves as an effective reduced-order model for chaotic systems.
• This method accurately predicts system dynamics and infers crucial stability properties from data.
• CAE-ESN offers a powerful tool for analyzing complex, chaotic physical systems.