Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds

TL;DR

This paper introduces two data-driven quadratic manifold methods for symplectic model reduction of Hamiltonian systems, improving accuracy and efficiency over traditional linear approaches, especially for wave-type problems.

Contribution

The paper develops two novel quadratic manifold-based symplectic reduction methods that better capture low-dimensional structures and outperform linear models in accuracy and extrapolation.

Findings

More accurate predictions than linear models.

Effective in extrapolating beyond training data.

Better representation of intrinsic low-dimensionality.

Abstract

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov $N$-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models.