Learning physics-based reduced-order models from data using nonlinear manifolds.

Area of Science:

• Dynamical Systems and Control Theory
• Machine Learning and Data Science
• Scientific Computing

Background:

• Reduced-order modeling (ROM) is crucial for simulating complex dynamical systems efficiently.
• Traditional ROM methods often rely on linear subspace approximations, limiting accuracy for nonlinear systems.
• Learning nonlinear structures in data is a key challenge in modern scientific modeling.

Purpose of the Study:

• To develop a novel method for learning reduced-order models of dynamical systems.
• To leverage nonlinear manifolds for improved model accuracy and generalizability.
• To demonstrate the effectiveness of the proposed approach over linear ROM techniques.

Main Methods:

• Learning nonlinear manifolds by identifying nonlinear structure in data via general representation learning.
• Utilizing embeddings of low-order polynomial form to drive the manifold learning process.
• Projecting onto the nonlinear manifold to reveal the reduced-space system's algebraic structure.
• Inferring reduced-order model matrix operators from data using operator inference.

Main Results:

• The methodology successfully captures the underlying nonlinear dynamics of the systems studied.
• Numerical experiments show significant increases in accuracy compared to linear subspace approximation methods.
• The approach demonstrates broad generalizability across various nonlinear problems.

Conclusions:

• The proposed nonlinear manifold-based method offers a more accurate and generalizable approach to reduced-order modeling.
• This technique advances the field of dynamical systems modeling by effectively handling nonlinearities.
• The findings pave the way for more efficient and reliable simulations of complex physical phenomena.