Structure-Preserving Discretization and Model Reduction for Energy-Based Models

TL;DR

This paper develops discretization and model reduction techniques for energy-based models, ensuring dissipation-preserving schemes applicable to systems like port-Hamiltonian and gradient flow models.

Contribution

It introduces a systematic framework combining discretization and reduction methods that preserve dissipation properties in energy-based models.

Findings

Numerical results demonstrate the effectiveness of the proposed schemes.

The approach applies to nonlinear circuit models and the Cahn-Hilliard equation.

Schemes maintain a discrete dissipation inequality.

Abstract

We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our framework combines existing ideas from the literature and systematically addresses temporal discretization, spatial discretization, and model order reduction, ensuring that all resulting schemes are dissipation-preserving in the sense of a discrete dissipation inequality. For this, we use a Petrov-Galerkin ansatz together with appropriate projections. Numerical results for a nonlinear circuit model and the Cahn-Hilliard equation illustrate the effectiveness of the approach.