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。