Variational Principle and Variational Integrators for Neural Symplectic Forms

In this study, we investigate the variational principle for neural symplectic forms, thereby designing the variational integrators for this model. In recent years, neural networks models for physical phenomena have been attracting much attention. In particular, the neural symplectic form is a method that can model general Hamiltonian systems, which are not necessary in the canonical form. In this paper, we make the following two contributions regarding this model. Firstly, we show that this model is derived from a variational principle and hence admits the Noether theorem.Secondly, when the trained models are used for simulations, they must be discretized using numerical integrators; however, unless carefully designed, numerical integrators destroy physical laws.