Learning nonlinear projections for reduced-order modeling of dynamical systems using constrained autoencoders.

Area of Science:

• Dynamical Systems and Control Theory
• Machine Learning for Scientific Modeling
• Fluid Dynamics

Background:

• Reduced-order modeling (ROM) approximates nonlinear dynamical systems using low-dimensional manifolds.
• Current ROMs excel in post-transient regimes but struggle with transient dynamics due to fast dynamics and nonnormal sensitivity.
• Accurate modeling of transient dynamics is crucial for real-time control and forecasting.

Purpose of the Study:

• To develop a new framework for nonlinear projections that accurately captures transient dynamics.
• To address limitations of existing ROMs in modeling complex system behaviors.
• To enable improved real-time control and forecasting capabilities.

Main Methods:

• Introduced constrained autoencoder neural networks for learning both manifolds and projection fibers.
• Employed invertible activation functions and biorthogonal weight matrices for encoder-decoder consistency.
• Developed dynamics-aware cost functions to learn oblique projection fibers accounting for fast dynamics and nonnormality.
• Utilized a three-state vortex shedding model for case study analysis.

Main Results:

• Demonstrated the ability of the proposed nonlinear projection framework to capture transient dynamics.
• Showcased the effectiveness of dynamics-aware cost functions in handling nonnormal effects.
• Validated the approach on a fluid dynamics case study with an analytically computed slow manifold.
• Proposed techniques for computationally efficient ROM construction, including sparsity promotion on the Grassmann manifold.

Conclusions:

• The developed nonlinear projection framework effectively models transient dynamics in complex systems.
• This approach enhances the applicability of reduced-order modeling for real-time control and forecasting.
• Future work can extend these methods to high-dimensional systems for broader scientific applications.