This paper presents a method for learning Hamiltonian dynamics from a limited set of data points. The Hamiltonian vector field is found by regularized optimization over a reproducing kernel Hilbert space of vector fields that are inherently Hamiltonian, and where the vector field is required to be odd or even. This is done with a symplectic kernel, and it is shown how this symplectic kernel can be modified to be odd or even. The performance of the method is validated in simulations for two Hamiltonian systems. It is shown that the learned dynamics are Hamiltonian, and that the learned Hamiltonian vector field can be prescribed to be odd or even.

翻译：本文提出了一种从有限数据点中学习哈密顿动力学的方法。通过在本质上为哈密顿的向量场的再生核希尔伯特空间中进行正则化优化，并限定向量场须具有奇偶性，从而求得哈密顿向量场。该方法采用辛核实现，并展示了如何对辛核进行修正以使其具备奇偶性。通过在两个哈密顿系统上的仿真验证了该方法的性能，结果表明：学习得到的动力学具有哈密顿结构，且所学习的哈密顿向量场能够被预设为奇函数或偶函数。