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

Dynamical and Optimization Trade-offs of Levi–Civita Coordinates for Learned Close-Encounter Dynamics
Abhishek Shankar
arXiv:2607.20235v1 Announce Type: cross
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 $2.1times10^{-5}$ 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 $3times10^{-5}$, 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.arXiv:2607.20235v1 Announce Type: cross
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 $2.1times10^{-5}$ 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 $3times10^{-5}$, 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.