Tensor parametric Hamiltonian operator inference
Arjun Vijaywargiya, Shane A. McQuarrie, Anthony Gruber
TL;DR
The paper tackles parametric model reduction for semi-discrete PDEs with Hamiltonian structure by introducing a tensorized, non-intrusive OpInf framework that encodes affine parameter dependence as $A(\\mu) = \mathbf{T} \\mu'$ and learns reduced operators via convex optimization. It unifies prior parametric OpInf approaches under a tensor contraction formulation and extends the method to preserve Hamiltonian structure by enforcing symmetry constraints on the learned tensor, yielding energy-conserving and symplectic reduced dynamics. The resulting structure-preserving ROMs are demonstrated on a heat equation with variable diffusion and a Hamiltonian wave equation with variable wave speed, showing competitive accuracy and stability, with clear distinctions between symmetric (structure-preserving) and non-symmetric OpInf variants. This approach offers a concise, implementable path toward nonintrusive, parametric, structure-preserving surrogates for complex physical systems relevant to predictive science and digital-twin applications.
Abstract
This work presents a tensorial approach to constructing data-driven reduced-order models corresponding to semi-discrete partial differential equations with canonical Hamiltonian structure. By expressing parameter-varying operators with affine dependence as contractions of a generalized parameter vector against a constant tensor, this method leverages the operator inference framework to capture parametric dependence in the learned reduced-order model via the solution to a convex, least-squares optimization problem. This leads to a concise and straightforward implementation which compactifies previous parametric operator inference approaches and directly extends to learning parametric operators with symmetry constraints, a key feature required for constructing structure-preserving surrogates of Hamiltonian systems. The proposed approach is demonstrated on both a (non-Hamiltonian) heat equation with variable diffusion coefficient as well as a Hamiltonian wave equation with variable wave speed.