We present a framework for learning Hamiltonian systems using data. This work is based on the lifting hypothesis, which posits that nonlinear Hamiltonian systems can be written as nonlinear systems with cubic Hamiltonians. By leveraging this, we obtain quadratic dynamics that are Hamiltonian in a transformed coordinate system. To that end, for given generalized position and momentum data, we propose a methodology to learn quadratic dynamical systems, enforcing the Hamiltonian structure in combination with a symplectic auto-encoder. The enforced Hamiltonian structure exhibits long-term stability of the system, while the cubic Hamiltonian function provides relatively low model complexity. For low-dimensional data, we determine a higher-order transformed coordinate system, whereas, for high-dimensional data, we find a lower-order coordinate system with the desired properties. We demonstrate the proposed methodology by means of both low-dimensional and high-dimensional nonlinear Hamiltonian systems.

使用数据学习Hamilton系统的框架，基于lifting假设，将非线性Hamilton系统用具有三次Hamilton量的非线性系统表示，并通过强制施加Hamilton结构和辛自编码器来学习二次动力系统，实现了系统的长期稳定性和相对较低的模型复杂度。

非线性哈密顿系统的基于数据的二次辛表示辨识