Adaptive data-driven reduced-order models for reciprocal MIMO port-Hamiltonian dynamical systems

Abstract

Port-Hamiltonian (pH) systems provide a general framework for modeling a broad class of physical dynamical systems, including systems with dissipation. Constructing accurate and robust data-driven reduced-order models for general pH systems is challenging. In this work, we focus on an important subclass of pH systems satisfying reciprocity, a property arising naturally in a variety of models. Rather than seeking to preserve the full pH structure, we exploit the additional symmetry provided by reciprocity to develop an efficient data-driven reduced-order modeling approach. We show that every minimal realization of a reciprocal system is self-adjoint with respect to a generally indefinite inner product and can, up to a similarity transformation preserving the pH structure, be represented in a Krein space with a signature inner product. This observation motivates a Galerkin projection onto a tangential rational Krylov subspace with respect to the corresponding indefinite inner product. Although the resulting projection involves full-order state solutions and the unknown inner product, we show that its reduced mass and stiffness matrices can be constructed directly from transfer-function data through a Loewner formulation. We then propose a greedy data-driven procedure that adaptively selects both interpolation frequencies and tangential directions. At every iteration, the frequency is chosen according to the largest transfer-function approximation error, while the tangential direction is selected from a dominant singular direction of the corresponding matrix-valued residual. Numerical experiments for wave propagation in lossy media demonstrate improved approximation efficiency compared with block adaptive interpolation. We further apply the method to spectral mass gauging of a three-dimensional propellant tank to provide an accurate estimate of the tank volume through Weyl's law.