We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules. In particular, we define two classes of SympNets: the LA-SympNets composed of linear and activation modules, and the G-SympNets composed of gradient modules. Correspondingly, we prove two new universal approximation theorems that demonstrate that SympNets can approximate arbitrary symplectic maps based on appropriate activation functions. We then perform several experiments including the pendulum, double pendulum and three-body problems to investigate the expressivity and the generalization ability of SympNets. The simulation results show that even very small size SympNets can generalize well, and are able to handle both separable and non-separable Hamiltonian systems with data points resulting from short or long time steps. In all the test cases, SympNets outperform the baseline models, and are much faster in training and prediction. We also develop an extended version of SympNets to learn the dynamics from irregularly sampled data. This extended version of SympNets can be thought of as a universal model representing the solution to an arbitrary Hamiltonian system.

我们提出了新的辛网络（SympNets），用于基于线性，激活和梯度模块的组合从数据中识别哈密顿系统。特别是，我们定义了两类SympNet：由线性和激活模块组成的LA-SympNet，以及由梯度模块组成的G-SympNet。相应地，我们证明了两个新的通用逼近定理，它们证明SympNets可以基于适当的激活函数逼近任意辛辛映射。然后我们进行包括摆，双摆和三体问题在内的几个实验，以研究SympNets的表达性和泛化能力。仿真结果表明，即使很小的SympNets都能很好地推广，并能够处理具有短时间或长时间步长的数据点的可分离和不可分离的汉密尔顿系统。在所有测试案例中，SympNets均优于基准模型，并且在训练和预测方面要快得多。我们还开发了SympNets的扩展版本，以从不规则采样的数据中学习动态。可以将SympNets的扩展版本视为代表任意哈密顿系统解决方案的通用模型。