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.

翻译：本文提出了一种利用数据学习哈密顿系统的框架。本工作基于提升假设——即非线性哈密顿系统可表示为具有三次哈密顿函数的非线性系统。通过利用这一性质，我们获得了在变换坐标系中保持哈密顿结构的二次动力学。为此，针对给定的广义位置和动量数据，我们提出了一种学习二次动力系统的方法，通过结合辛自编码器强制施加哈密顿结构。所强制的哈密顿结构保证了系统的长期稳定性，而三次哈密顿函数则提供了相对较低的模型复杂度。对于低维数据，我们确定了一个高阶变换坐标系；而对于高维数据，则寻找具有所需性质的低阶坐标系。我们通过低维和高维非线性哈密顿系统实例验证了所提方法的有效性。