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Many physical phenomena are described by Hamiltonian mechanics using anenergy function (the Hamiltonian). Recently, the Hamiltonian neural network,which approximates the Hamiltonian as a neural network, and its extensions haveattracted much attention. This is a very powerful method, but its use intheoretical studies remains limited. In this study, by combining thestatistical learning theory and Kolmogorov-Arnold-Moser (KAM) theory, weprovide a theoretical analysis of the behavior of Hamiltonian neural networkswhen the learning error is not completely zero. A Hamiltonian neural networkwith non-zero errors can be considered as a perturbation from the truedynamics, and the perturbation theory of the Hamilton equation is widely knownas the KAM theory. To apply the KAM theory, we provide a generalization errorbound for Hamiltonian neural networks by deriving an estimate of the coveringnumber of the gradient of the multi-layer perceptron, which is the keyingredient of the model. This error bound gives an $L^\infty$ bound on theHamiltonian that is required in the application of the KAM theory.

KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss