Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics

• URL: http://arxiv.org/abs/2607.20235v1
• Date: Wed, 22 Jul 2026 14:56:15 GMT
• Title: Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics
• Abstract summary: We show that a Levi--Civita Hamiltonian splitting holds the maximum relative energy error near $2.1times10-5$.<n>Regularized models produce finite rollouts in $40$ runs versus $0/40$ for Cartesian.<n>Four neural residual objectives fail to approach the analytic result.<n>This is a controlled falsification-plus-trade-off study, not a solution to learned close-encounter dynamics.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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