提出可自适应时间步长的哈密顿神经网络，突破传统方法对等距采样的依赖。
Time-adaptive HénonNets for separable Hamiltonian systems
- 设计基于辛结构的T-HénonNets，支持非均匀时间采样。
- 在可分哈密顿系统上实现高精度长期积分，误差随时间增长缓慢。
- 适用于物理模拟中采样不规则的实际场景，如天体动力学。
测量数据常以非均匀时间间隔采样，哈密顿系统亦如此。然而现有机器学习方法如SympNets和HénonNets仍需固定步长训练数据。本文提出T-HénonNets，一种基于辛结构设计的新型神经网络，可处理自适应时间步长。该架构进一步扩展至非自治哈密顿系统，并为可分哈密顿系统提供了通用逼近定理；同时分析了为何难以应用于不可分系统。通过多种数值实验验证了其理论逼近能力，结果显示在长期积分中保持低误差累积。
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 [1] and HénonNets [2] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets called TSympNets is introduced in [3]. The aim of this work is to do a similar extension for HénonNets. We propose a novel neural network architecture called T-HénonNets, which is symplectic by design and can handle adaptive time steps. We also extend the T-HénonNet architecture to non-autonomous Hamiltonian systems. Additionally, we provide universal approximation theorems for both new architectures for separable Hamiltonian systems and discuss why it is difficult to handle non-separable Hamiltonian systems with the proposed methods. To investigate these theoretical approximation capabilities, we perform different numerical experiments.
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