SupplementaryMaterial: Appendices

Symplectic integrators arethe numerical integrators thatpreservethisconservation law;hence, theycanbeinasense considered as adiscrete Hamiltonian system that is an approximation to the target Hamiltonian system. As shown above, a discrete gradient is defined in Definition 1. However,most oftheexisting discrete gradients require explicit representation of the Hamiltonian; hence, they are not available for neural networks. An exception is the Ito-Abe method[24] Hence, the proposed automatic discrete differentiation algorithm isindispensable for practical application of the discrete gradient methodforneuralnetworks. Seealso [17,22]. The target equations for this study are the differential equations with acertain geometric structure. The typical examples of the manifolds with such a2-tensor are the Riemannian manifold [4]and thesymplectic manifold [29].