Hamilton流映射的辛高斯过程回归
Symplectic Gaussian Process Regression of Hamiltonian Flow Maps
摘要：
我们提出了一种方法来构建适当的和有效的仿真哈密顿流映射。预期的未来应用是在加速器和磁等离子体约束配置的快速带电粒子的长期跟踪。该方法是基于多输出高斯过程回归分散的训练数据。为了获得长期稳定性的辛属性是通过选择矩阵值协方差函数。基于以前的工作样条插值，我们观察到的衍生物的生成函数的一个典型的变换。乘积核产生精确的隐式方法，而和核产生快速的显式方法。这两种方法在数值积分方面都对应于辛欧拉方法。这些方法被施加到摆和Hénon-Heiles系统和结果相比，一个对称回归正交多项式。在小映射时间的限制下，可以用生成函数的一部分来识别哈密顿函数，从而从观测到的系统演化的时间序列数据中学习。除了辛映射的隐式核和谱回归的可比性能外，与现有方法相比，我们在学习Hamilton函数方面的性能有了大幅提高。
英文摘要：
We present an approach to construct appropriate and efficient emulators for Hamiltonian flow maps. Intended future applications are long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process regression on scattered training data. To obtain long-term stability the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both correspond to a symplectic Euler method in terms of numerical integration. These methods are applied to the pendulum and the Hénon-Heiles system and results compared to an symmetric regression with orthogonal polynomials. In the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. Besides comparable performance of implicit kernel and spectral regression for symplectic maps, we demonstrate a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.
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