A million asteroids no longer lean on soft constraints: symplectic graph neural ODEs cut position error to 1.8×10⁻⁴ AU

Orbit reconstruction for asteroid swarms across observation gaps no longer relies on soft conservation penalties; the Hamiltonian itself constrains the evolution. Earlier physics-informed neural networks and data-driven neural ODEs wrote conservation laws as soft penalty terms in the loss, which produced phase drift, spurious orbital expansion and energy dissipation outside observed coverage. On a simulated swarm of one million bodies spanning the Main Belt, Jupiter Trojans L4/L5, near-Earth asteroids and Centaurs, the method reaches a mean position error of 1.8005×10⁻⁴ AU, energy drift of 8.42×10⁻⁵ and phase-space dissipation of 3.31×10⁻⁷, compared against eight baselines including soft PINNs, HNN-RK4, SympNet, EG-NODE and PINO-FNO.

The network writes the analytic part of multi-body gravity directly into the forward pass and lets the neural network learn only the remaining higher-order perturbations. Earlier approaches either learned the gravitational field wholesale from data or added conservation laws as an extra training objective. The total Hamiltonian combines kinetic energy, a multi-body Newtonian potential and a learned residual potential; the residual is parameterized by an SE(3)-equivariant graph neural network whose edge features are relative displacement, relative velocity and the ratio of displacement to sphere-of-influence radius.

Message passing between bodies is automatically modulated by planetary sphere-of-influence radii, so locally bound orbits and free hyperbolic flybys are distinguished within one graph. Earlier graph networks used uniform edge weights for all body pairs and could not separate bound states near a planet from free flight in interplanetary space. Routing weights come from the Laplace sphere-of-influence radius R_SOI = a(m/M)^(2/5) passed through a sigmoid; Jupiter's radius is about 0.354 AU and Earth's about 0.0062 AU, with weights approaching 1 inside Jupiter's sphere and 0 in interplanetary space.

Time stepping uses a symplectic implicit midpoint scheme, keeping energy drift bounded over any integration interval rather than accumulating. Non-symplectic integrators such as explicit fourth-order Runge-Kutta break phase-space volume conservation and cause long-term energy dissipation. The implicit midpoint rule belongs to the symplectic Runge-Kutta family with an energy drift bound of O(Δt²); on the million-body batch the method completes one evaluation in 1.48 s on an ordinary workstation GPU using 2.4 GB of VRAM.

The method handles small-body orbits under multi-body gravity, 1PN Schwarzschild precession and solar radiation pressure in a barycentric inertial frame. Validation covers a one-million-body synthetic swarm across four orbital families and NASA JPL Horizons ephemerides for 99942 Apophis from January 1 to July 1, 2029. It is aimed at researchers and engineers working on orbit determination, planetary defense and scientific machine learning. Future work will extend the framework to Yarkovsky thermal recoil accelerations and 2PN post-Newtonian expansions.

In the synthetic swarm, observation degradation combines a contiguous 35% blackout, 30% random dropout and Gaussian noise with σ=1.5×10⁻³ AU; the Apophis case uses a contiguous 60-day blackout covering 33.0% of the arc. Jupiter Trojans and Centaurs show a maximum error of 7.8764×10⁻⁴ AU, higher than the Main Belt and near-Earth asteroids, attributed to stronger nonlinear multi-body interactions inside Jupiter's sphere of influence.