这篇论文用实验数据说明了Levi-Civita坐标在神经哈密顿动力学中能稳定高偏心率轨迹，但优化条件更差，适合想了解正则化坐标利弊的研究者。
本文系统对比笛卡尔坐标与Levi-Civita坐标在平滑四极势扰动开普勒问题中的表现。使用Levi-Civita哈密顿分裂，最大相对能量误差保持2.1×10⁻⁵，偏心率e=0.99时笛卡尔分裂不稳定。在固定壳构造下，高偏心率测试中正则化模型40/40次生成有限轨迹，笛卡尔模型0/40。正则化残差存在严重病态问题，正交化可恢复基线拟合，但小型MLP的轨迹误差仍为O(1)。这是一项受控的证伪加权衡研究，未解决精确神经残差学习。
Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics
Classical regularization removes the binary-collision singularity from the Kepler problem, but its value as a representation for learned Hamiltonian dynamics has not been systematically isolated. We compare Cartesian and planar Levi--Civita formulations of a perturbed Kepler system with a smooth quadrupole potential. With the perturbation supplied analytically, a Levi--Civita Hamiltonian splitting holds the maximum relative energy error near $2.1\times10^{-5}$ through eccentricity $e=0.99$, while the Cartesian splitting becomes unstable. This advantage persists at matched physical horizon and force-evaluation budget, where the regularized baseline is $3\times10^{-5}$, about $4.7$--$8.3$ orders of magnitude below the Cartesian arm depending on eccentricity. In held-out high-eccentricity tests with matched sampling, regularized models produce finite rollouts in $40/40$ runs versus $0/40$ for Cartesian. However, the fixed-shell construction supplies the regularized model with the exact initial orbit energy, and survival still carries $\mathcal{O}(1)$ energy error. Four neural residual objectives fail to approach the analytic result. Exact-feature controls show that the regularized residual is a four-monomial degree-6 polynomial that a direct least-squares solve fits to the baseline. The remaining exact-feature gap is due to severe raw-basis ill-conditioning: orthogonalization restores baseline fitting for L-BFGS in two iterations. Small MLPs remain at $\mathcal{O}(1)$ rollout error even after gauge symmetrization. Levi--Civita coordinates therefore improve dynamical conditioning while worsening raw-basis optimization conditioning; accurate neural residual learning remains unresolved. This is a controlled falsification-plus-trade-off study, not a solution to learned close-encounter dynamics.