Deep learning to discover and predict dynamics on an inertial manifold

Area of Science:

• Computational physics
• Dynamical systems theory
• Machine learning applications

Background:

• Chaotic dynamics are complex and challenging to model accurately.
• Inertial manifolds (IMs) provide reduced-order representations for dissipative systems.
• Existing dimension reduction techniques often struggle with capturing essential dynamics.

Purpose of the Study:

• To develop a data-driven framework for representing chaotic dynamics on an inertial manifold (IM).
• To apply this framework to the Kuramoto-Sivashinsky equation, a model for chaotic dynamics.
• To improve the accuracy and efficiency of modeling complex dynamical systems.

Main Methods:

• A hybrid dimension reduction technique combining linear and nonlinear (neural network) methods.
• Transformation between full state space coordinates and IM coordinates using neural networks.
• Time evolution prediction on the IM using additional neural networks.
• Incorporation of physical constraints like translation invariance and energy conservation.

Main Results:

• The developed framework accurately represents chaotic dynamics on the IM.
• The hybrid method substantially outperforms traditional linear dimension reduction.
• Key dynamic and statistical features of the attractor are well reproduced.
• The approach effectively captures translation invariance and energy conservation.

Conclusions:

• The data-driven framework offers a powerful new tool for analyzing chaotic dynamics on IMs.
• Neural network-based dimension reduction provides significant advantages over linear methods.
• This formalism enables more faithful and efficient modeling of complex systems.