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

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.