  Many Hamiltonian systems can be recast in multi-symplectic form. We develop a
reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential
equations (PDEs) that preserves the global energy. The full-order solutions are
obtained by finite difference discretization in space and the global energy
preserving average vector field (AVF) method. The ROM is constructed in the
same way as the full-order model (FOM) applying proper orthogonal decomposition
(POD) with the Galerkin projection. The reduced-order system has the same
structure as the FOM, and preserves the discrete reduced global energy.
Applying the discrete empirical interpolation method (DEIM), the reduced-order
solutions are computed efficiently in the online stage. A priori error bound is
derived for the DEIM approximation to the nonlinear Hamiltonian. The accuracy
and computational efficiency of the ROMs are demonstrated for the Korteweg de
Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear
Schr{\"o}dinger (NLS) equation in multi-symplectic form. Preservation of the
reduced energies shows that the reduced-order solutions ensure the long-term
stability of the solutions.
