Data-driven guessing and gluing of unstable periodic orbits

Area of Science:

• * Dynamical systems theory
• * Computational physics
• * Fluid dynamics and turbulence

Background:

• * Unstable periodic orbits (UPOs) are fundamental to understanding spatiotemporal chaos and turbulence.
• * Traditional methods for finding UPOs rely on generating initial guesses for loop convergence algorithms, which is computationally demanding and often limited to simpler orbits.
• * The high dimensionality of fluid flow state spaces makes constructing suitable initial guesses difficult.

Purpose of the Study:

• * To develop a novel, data-driven approach for generating effective initial guesses for loop convergence algorithms.
• * To leverage low-dimensional representations of complex dynamics for UPO discovery.
• * To improve the efficiency and applicability of finding UPOs in chaotic systems.

Main Methods:

• * Utilized an autoencoder to learn a low-dimensional latent space representation of the one-dimensional Kuramoto-Sivashinsky equation.
• * Constructed initial guesses (loops) in the latent space using Proper Orthogonal Decomposition (POD) modes with random periodic coefficients.
• * Decoded these latent space loops back into the physical space for use with variational convergence algorithms.

Main Results:

• * The autoencoder successfully captured the low-dimensional chaotic attractor of the system.
• * Generated loops in the latent space served as effective initial guesses, enabling rapid convergence to UPOs.
• * A 'gluing' procedure in the latent space successfully generated guesses for longer UPOs, suggesting a UPO hierarchy.

Conclusions:

• * The proposed data-driven method significantly enhances the ability to find UPOs in chaotic systems.
• * The low-dimensional latent space approach offers a powerful alternative to traditional methods for generating initial guesses.
• * The findings suggest a hierarchical structure of UPOs, where longer orbits shadow sequences of shorter ones.