发表机构
Ward Melville High School; Michigan State University; Brookhaven National Lab（沃德·梅维尔高中; 密歇根州立大学; 布鲁克海文国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究利用辛神经网络学习非线性映射的正规形变换，在保持辛结构的同时实现长期跟踪，并在四维McMillan映射上验证了其精度与稳定性。
AI 中文摘要
研究环形加速器中的非线性现象（如共振、混沌运动和动态孔径）需要进行长期多圈跟踪。通用神经网络不保持哈密顿动力学的辛结构，因此其误差可能在多圈中累积增长。辛神经网络（SympNets）通过其架构而非损失惩罚来保证辛性。在本工作中，我们使用SympNet从跟踪数据中学习非线性映射的正规形变换。在学习坐标下，动力学变为每个模式的旋转，其振幅依赖的相移由第二个网络给出。我们在一个四维McMillan型映射上演示了该方法。模型再现了单圈动力学，学习到的相移沿每条轨迹保持恒定，误差在$\sim 10^{-5}$弧度以内，且在小到中等振幅下，轨道在学习坐标中接近圆形。在大振幅下，学习到的轨道明显发散，导致这种退化的原因尚未确定。该方法朝着存储环中晶格分析和在线束流动力学应用的快速、结构保持代理模型迈出了一步。
英文摘要
Long-term multi-turn tracking is required to study nonlinear phenomena in circular accelerators, such as resonances, chaotic motion, and the dynamic aperture. General-purpose neural networks do not preserve the symplectic structure of Hamiltonian dynamics, so their errors can grow over many turns. Symplectic neural networks (SympNets) guarantee symplecticity through their architecture rather than through a loss penalty. In this work, we use a SympNet to learn the normal-form transformation of a nonlinear map from tracking data. In the learned coordinates, the dynamics become a rotation of each mode, whose amplitude-dependent phase advance is given by a second network. We demonstrate the method on a four-dimensional McMillan-type map. The model reproduces the one-turn dynamics, the learned phase advances stay constant along each trajectory to within $\sim 10^{-5}$~rad, and orbits become close to circles in the learned coordinates at small and moderate amplitude. At large amplitude the learned orbits spread noticeably, and the cause of this degradation is not yet established. The approach is a step toward fast, structure-preserving surrogate models for lattice analysis and online beam-dynamics applications in storage rings.
