提出可自适应时间步长的辛神经网络，提升非自治哈密顿系统建模精度
Time-adaptive SympNets for separable Hamiltonian systems
- 设计时间自适应架构，支持不规则采样数据下的辛积分学习
- 证明可逼近可分哈密顿系统，但不可扩展至不可分系统
- 修正了对称映射近似理论的关键错误，增强方法可靠性
测量数据常以非等距时间间隔采样，哈密顿系统亦然。现有机器学习方法如SympNets和HénonNets仍需固定步长训练数据。为学习时间自适应的辛积分器，文献[20]提出TSympNets，本文对其架构进行改进并扩展至非自治哈密顿系统。此前TSympNets的逼近性能未知，本文首次为可分哈密顿系统提供通用逼近定理，并证明无法推广至不可分系统。通过数值实验验证理论结果，同时修正了文献[25, Theorem 2]中关于对称映射逼近的关键证明错误，该错误影响对称机器学习方法的理论基础。
原文摘要 · Abstract (English)
Measurement data is often sampled irregularly i.e. not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [20] and HénonNets [4] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets, which we call TSympNets, was introduced in [20]. We adapt the architecture of TSympNets and extend them to non-autonomous Hamiltonian systems. So far the approximation qualities of TSympNets were unknown. We close this gap by providing a universal approximation theorem for separable Hamiltonian systems and show that it is not possible to extend it to non-separable Hamiltonian systems. To investigate these theoretical approximation capabilities, we perform different numerical experiments. Furthermore we fix a mistake in a proof of a substantial theorem [25, Theorem 2] for the approximation of symplectic maps in general, but specifically for symplectic machine learning methods.
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