  Many physical phenomena are described by Hamiltonian mechanics using an
energy function (the Hamiltonian). Recently, the Hamiltonian neural network,
which approximates the Hamiltonian as a neural network, and its extensions have
attracted much attention. This is a very powerful method, but its use in
theoretical studies remains limited. In this study, by combining the
statistical learning theory and Kolmogorov-Arnold-Moser (KAM) theory, we
provide a theoretical analysis of the behavior of Hamiltonian neural networks
when the learning error is not completely zero. A Hamiltonian neural network
with non-zero errors can be considered as a perturbation from the true
dynamics, and the perturbation theory of the Hamilton equation is widely known
as the KAM theory. To apply the KAM theory, we provide a generalization error
bound for Hamiltonian neural networks by deriving an estimate of the covering
number of the gradient of the multi-layer perceptron, which is the key
ingredient of the model. This error bound gives an $L^\infty$ bound on the
Hamiltonian that is required in the application of the KAM theory.
