  HNets is a class of neural networks on grounds of physical prior for learning
Hamiltonian systems. This paper explains the influences of different
integrators as hyper-parameters on the HNets through error analysis. If we
define the network target as the map with zero empirical loss on arbitrary
training data, then the non-symplectic integrators cannot guarantee the
existence of the network targets of HNets. We introduce the inverse modified
equations for HNets and prove that the HNets based on symplectic integrators
possess network targets and the differences between the network targets and the
original Hamiltonians depend on the accuracy orders of the integrators. Our
numerical experiments show that the phase flows of the Hamiltonian systems
obtained by symplectic HNets do not exactly preserve the original Hamiltonians,
but preserve the network targets calculated; the loss of the network target for
the training data and the test data is much less than the loss of the original
Hamiltonian; the symplectic HNets have more powerful generalization ability and
higher accuracy than the non-symplectic HNets in addressing predicting issues.
Thus, the symplectic integrators are of critical importance for HNets.
