Estimating the Hamiltonian Hessian or selecting an alternative test basis

Improve the estimation of the Hamiltonian Hessian matrix for Petrov–Galerkin operator inference with unknown Hamiltonian structure, or identify alternative test-basis choices that avoid reliance on an accurately estimated Hessian.

Background

The structure-preserving Petrov–Galerkin operator-inference method uses the test basis defined by the Hessian of the Hamiltonian. When this Hessian is unknown, the paper estimates it from Hamiltonian measurements, but the numerical experiments show that the estimate can fail to converge and lead to poor reduced models. The authors therefore leave open both the improvement of Hessian estimation and the development of alternative test spaces that do not depend on this matrix.

References

We conclude, that PG-OpInf demonstrates strong potential. However, in practical applications when the Hamiltonian Hessian is unknown, our current approach to determine~$W$ is not yet fully satisfactory. Further research is needed to improve the estimation of~$Q$ or to identify alternative choices for $W$, which remains an active area of investigation; cf..

— Petrov-Galerkin operator inference with application to stability-encouraging identification  (2610.01295 - Rettberg et al., 1 Oct 2026) in Section 5.3, subsection “Inferring the Hamiltonian and the pH One-shot Identification”