用勒维-奇维塔坐标提升近心点动力学学习稳定性，但优化难度增加。
Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics
- 采用勒维-奇维塔坐标重构引力系统，避免碰撞奇点
- 在偏心率0.99下能量误差仅2.1×10⁻⁵，远优于笛卡尔坐标
- 适合研究高偏心轨道的神经哈密顿建模与数值稳定性
经典正则化可消除二体问题中的碰撞奇点，但其在学习哈密顿动力学中的作用尚未系统验证。本文对比了受四极势扰动的开普勒系统在笛卡尔坐标与平面勒维-奇维塔坐标下的表现。在解析扰动条件下，勒维-奇维塔哈密顿分裂在偏心率e=0.99时保持最大相对能量误差约2.1×10⁻⁵，而笛卡尔分裂出现不稳定性。在相同物理时域和力评估预算下，正则化基线误差为3×10⁻⁵，比笛卡尔方案低4.7至8.3个数量级。在高偏心率外推测试中，正则化模型在40/40次运行中保持有限轨迹，而笛卡尔模型全失败。尽管正则化模型初始能量精确，仍存在𝒪(1)能量误差。四种神经残差目标均无法逼近解析解。精确特征分析表明，正则化残差为六次四项单项式，直接最小二乘法可拟合基准。剩余误差源于原始基底严重病态：正交化后L-BFGS两步即恢复基准拟合。小MLP即使经规范对称化，仍维持𝒪(1)轨迹误差。因此，勒维-奇维塔坐标改善动力学条件性，却恶化原始基优化条件性；神经残差学习的准确性仍未解决。本研究为受控的证伪与权衡实验，非近心点动力学学习的解决方案。
原文摘要 · Abstract (English)
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.
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