AI 中文总结
研究比较笛卡尔和列维 - 奇维塔公式用于受扰开普勒系统，发现列维 - 奇维塔坐标改善动力学条件但恶化原始基优化条件，在高偏心率测试中有优势，神经残差学习未解决，是关于近距离相遇动力学的证伪加权衡研究。
AI 中文摘要
经典正则化消除了开普勒问题中的二体碰撞奇点，但其作为学习哈密顿动力学表示的价值尚未得到系统分离。我们比较了具有光滑四极势的受扰开普勒系统的笛卡尔和平面列维 - 奇维塔公式。通过解析提供扰动，列维 - 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接近\(2.1×10^{-5}\)，而笛卡尔分裂变得不稳定。在匹配的物理视界和力评估预算下，正则化基线为\(3×10^{-5}\)，比笛卡尔方法低约\(4.7 - 8.3\)个数量级。在高偏心率测试中，正则化模型在\(40/40\)次运行中产生有限的展开，而笛卡尔方法为\(0/40\)。然而，固定壳结构给正则化模型提供了精确的初始轨道能量，生存仍带有\(\mathcal{O}(1)\)的能量误差。四个神经残差目标未能接近解析结果。精确特征控制表明正则化残差是一个四次六项式多项式，直接最小二乘法求解可拟合基线。剩余的精确特征差距是由于原始基严重病态：正交化在两次迭代中恢复了L - BFGS的基线拟合。即使经过规范对称化，小多层感知器仍保持\(\mathcal{O}(1)\)的展开误差。因此，列维 - 奇维塔坐标改善了动力学条件，但恶化了原始基优化条件；精确的神经残差学习仍未解决。这是一项受控的证伪加权衡研究，而非学习到的近距离相遇动力学的解决方案。
英文摘要
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.
Comments16 pages, 4 figures
