We present a framework for learning Hamiltonian systems using data. This work is based on a 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 weakly-enforced symplectic auto-encoder. The obtained 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-dimensional transformed coordinate system, whereas for high-dimensional data, we find a lower-dimensional 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：我们提出了一种利用数据学习哈密顿系统的框架。该工作基于提升假设，即非线性哈密顿系统可表示为具有三次哈密顿量的非线性系统。基于这一假设，我们得到了在变换坐标系中具有哈密顿结构的二次动力学系统。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学系统的方法，该方法在弱约束辛自编码器下强制保持哈密顿结构。所获得的哈密顿结构使系统具有长期稳定性，而三次哈密顿函数则实现了较低的模型复杂度。对于低维数据，我们确定了一个高维变换坐标系；而对于高维数据，则找到了一个具有所需属性的低维坐标系。我们通过低维和高维非线性哈密顿系统案例验证了所提方法。