Structure-Preserving Discretization and Model Reduction for Energy-Based Models
This work provides a systematic approach to preserving dissipation in numerical simulations of energy-based models, which is important for accurate long-term behavior in physics and engineering applications.
The authors develop a structure-preserving discretization and model reduction framework for energy-based models that ensures a discrete dissipation inequality. Numerical tests on a nonlinear circuit, Cahn-Hilliard equation, and doubly nonlinear parabolic equation demonstrate the method's effectiveness.
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, the Cahn--Hilliard equation, and a doubly nonlinear parabolic equation illustrate the effectiveness of the approach.