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We propose new symplectic networks (SympNets) for identifying Hamiltoniansystems from data based on a composition of linear, activation and gradientmodules. In particular, we define two classes of SympNets: the LA-SympNetscomposed of linear and activation modules, and the G-SympNets composed ofgradient modules. Correspondingly, we prove two new universal approximationtheorems that demonstrate that SympNets can approximate arbitrary symplecticmaps based on appropriate activation functions. We then perform severalexperiments including the pendulum, double pendulum and three-body problems toinvestigate the expressivity and the generalization ability of SympNets. Thesimulation results show that even very small size SympNets can generalize well,and are able to handle both separable and non-separable Hamiltonian systemswith data points resulting from short or long time steps. In all the testcases, SympNets outperform the baseline models, and are much faster in trainingand prediction. We also develop an extended version of SympNets to learn thedynamics from irregularly sampled data. This extended version of SympNets canbe thought of as a universal model representing the solution to an arbitraryHamiltonian system.

SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems