哈密顿神经网络在单摆与开普勒动力学上的匹配积分器评估
A matched-integrator evaluation of Hamiltonian neural networks on pendulum and Kepler dynamics
- African Institute for Mathematical Sciences (AIMS) Senegal（塞内加尔非洲数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究通过匹配积分器协议，对比哈密顿神经网络（HNN）与参数匹配前馈基线，在单摆、开普勒二体问题上验证了HNN在降低能量、轨迹漂移及提升物理一致性上的优势。
AI中文摘要：
哈密顿神经网络（Hamiltonian Neural Networks, HNNs）通过学习到的标量哈密顿量来参数化保守动力学，这提供了通用向量场神经网络所不具备的架构先验。我们在受控协议下评估该先验：将HNN与参数匹配的前馈基线在相同的RK4生成轨迹上训练，使用相同的中心差分导数目标与优化设置，并在推理阶段用相同的RK4方案积分，结果基于5个独立训练种子报告。在非线性单摆上，当T=100（约16个单摆周期）时，HNN将平均能量漂移降低42倍，平均轨迹MSE降低15.8倍；其能量漂移保持有界，且种子间变异性远低于标准网络基线。按能量分层的分析显示，当轨迹探索相空间中更非线性的区域时，两者的差异会更显著。作为额外诊断，我们检查了学习到的HNN的显式Störmer–Verlet式滚动，由于学习到的哈密顿量未被约束为可分离形式H(q,p)=T(p)+V(q)，速度Verlet的标准辛性保证无法直接适用。我们进一步将相同的匹配积分器协议应用于三维开普勒二体问题，HNN再次展现出比参数匹配基线更低的轨迹、能量和角动量漂移。这些实验对两种保守动力学系统中，哈密顿参数化如何影响长时程预测与物理一致性进行了受控研究。
英文摘要：
Hamiltonian Neural Networks (HNNs) parameterize conservative dynamics through a learned scalar Hamiltonian, providing an architectural prior that is absent from generic vector-field neural networks. We evaluate this prior under a controlled protocol in which an HNN and a parameter-matched feedforward baseline are trained on the same RK4-generated trajectories, use the same central-difference derivative targets and optimization settings, and are integrated at inference with the same RK4 scheme. Results are reported over five independent training seeds. On the nonlinear pendulum, the HNN reduces mean energy drift by 42-fold and mean trajectory MSE by 15.8-fold at T = 100, approximately 16 pendulum periods. Its energy drift also remains bounded and exhibits substantially lower seed-to-seed variability than the standard-network baseline. An energy-stratified analysis shows that the difference becomes more pronounced as trajectories explore more nonlinear regions of phase space. As an additional diagnostic, we examine an explicit Störmer--Verlet-style rollout of the learned HNN. Because the learned Hamiltonian is not constrained to the separable form H(q,p) = T(p) + V(q), the standard symplecticity guarantee of velocity Verlet does not directly apply. We further apply the same matched-integrator protocol to the three-dimensional Kepler two-body problem. The HNN again exhibits lower trajectory, energy, and angular-momentum drift than the parameter-matched baseline. These experiments provide a controlled study of how Hamiltonian parameterization affects long-horizon prediction and physical consistency across two conservative dynamical systems.
