  This work presents a nonintrusive physics-preserving method to learn
reduced-order models (ROMs) of canonical Hamiltonian systems. Traditional
intrusive projection-based model reduction approaches utilize symplectic
Galerkin projection to construct Hamiltonian ROMs by projecting Hamilton's
equations of the full model onto a symplectic subspace. This symplectic
projection requires complete knowledge about the full model operators and full
access to manipulate the computer code. In contrast, the proposed Hamiltonian
operator inference approach embeds the physics into the operator inference
framework to develop a data-driven model reduction method that preserves the
underlying symplectic structure. Our method exploits knowledge of the
Hamiltonian functional to define and parametrize a Hamiltonian ROM form which
can then be learned from data projected via symplectic projectors. The proposed
method is gray-box in that it utilizes knowledge of the Hamiltonian structure
at the partial differential equation level, as well as knowledge of spatially
local components in the system. However, it does not require access to computer
code, only data to learn the models. Our numerical results demonstrate
Hamiltonian operator inference on a linear wave equation, the cubic nonlinear
Schr\"{o}dinger equation, and a nonpolynomial sine-Gordon equation. Accurate
long-time predictions far outside the training time interval for nonlinear
examples illustrate the generalizability of our learned models.
