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Hard Symplectic Graph Neural ODEs for Scalable Trajectory Reconstruction in Perturbed Relativistic N-Body Celestial Swarms: Scaling to One Million Bodies with Empirical NASA Ephemerides

This paper introduces SympG-NODE, a structure-preserving Graph Neural ODE framework that leverages symplectic geometry and equivariant graph networks to achieve scalable, high-accuracy trajectory reconstruction for perturbed relativistic N-body celestial swarms of up to one million bodies while strictly conserving Hamiltonian invariants and overcoming the phase-drift limitations of standard Physics-Informed Neural Networks.

Original authors: Quan Van

Published 2026-10-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Quan Van

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The sky above us is not a static backdrop but a dynamic, crowded highway of moving stones. Astronomers have cataloged millions of these objects, from the vast belts of asteroids between Mars and Jupiter to the icy wanderers drifting on the solar system's outer edges. Keeping track of them is a matter of planetary safety; if a large rock were to veer off course and strike Earth, knowing its exact path years in advance is the only way to prevent a catastrophe. However, our view of this cosmic traffic is frequently interrupted. When an asteroid passes behind the Sun, or when a telescope is blinded by daylight or weather, the object disappears from our sensors for weeks or even months. During these dark gaps, the object continues to move, but we are left guessing its location. Traditional methods of predicting where it will reappear often fail when the data is missing for too long, causing the predicted path to drift wildly off course. This is because the mathematics used to fill in the blanks often forgets the fundamental rules that govern how energy and motion behave in space, leading to predictions that look plausible at first but become physically impossible over time.

A researcher named Quan Van has developed a new way to solve this problem, creating a computer system that learns the laws of motion so deeply that it never forgets them, even when the data goes silent. This system, called SympG-NODE, is designed to track the movements of up to one million celestial bodies at once, a scale that was previously impossible to handle with such precision. The core idea is to build a digital model that does not just guess the next position of an asteroid based on patterns, but instead calculates it by strictly adhering to the conservation of energy and the geometry of space. In the real world, if you know the position and speed of a planet, you can predict its future path forever without the prediction slowly drifting away from reality. Standard computer programs often lose this precision over long periods, accumulating tiny errors that eventually make the entire prediction useless. Van's system avoids this by hard-coding the rules of physics directly into its learning process, ensuring that the energy of the system remains constant, just as it does in nature.

The challenge Van faced was not just about one asteroid, but about the chaotic dance of a million of them. The solar system is a complex web of gravitational pulls. While we often think of the Sun as a stationary anchor, it actually wobbles slightly due to the massive gravity of Jupiter and Saturn. To get accurate results, the new system treats the entire solar system as a single, moving unit, calculating the position of every asteroid relative to the center of mass of the whole system rather than a fixed point. Furthermore, the system recognizes that an asteroid's behavior changes depending on where it is. When a rock is far out in space, it is dominated by the Sun's gravity. But when it swings close to a giant planet like Jupiter, the planet's gravity takes over, and the asteroid enters a temporary, local orbit. The new model uses a smart routing mechanism that automatically switches its focus between the Sun and the nearby planet, depending on how close the asteroid is to the planet's "sphere of influence." This allows the computer to handle the transition from a wide solar orbit to a tight planetary flyby without losing track of the object.

To test if this approach actually works, the researchers put it through two extreme trials. First, they used real data from NASA regarding a specific near-Earth asteroid named 99942 Apophis, which is expected to make a very close pass by Earth in 2029. They simulated a scenario where the asteroid was hidden from view for sixty days during its approach. When they asked the new system to fill in the missing path, it succeeded with remarkable accuracy, keeping the predicted position within a fraction of a mile of the true path. In contrast, other modern computer models, which rely on softer, less rigid rules, showed significant errors; for instance, the Soft-PINN model drifted by approximately 2.3 million miles, while even more advanced geometric baselines like HNN-RK4 still accumulated errors. The new system, however, maintained sub-millimeter accuracy relative to the true path, effectively losing the asteroid. The new system also correctly calculated the asteroid's speed and energy, ensuring that it remained on a bound path around the Sun rather than falsely predicting it would fly off into deep space.

The second test was a massive simulation involving one million artificial asteroids spread across different families in the solar system, including the main asteroid belt, the Trojan clusters near Jupiter, and the distant Centaurs. This test was designed to see if the system could scale up to handle the sheer volume of data required for a full census of the solar system. The system processed all one million objects in less than two seconds on a standard computer graphics card, a speed that would have been impossible for previous methods. It maintained high precision for every single object, keeping the error in position to less than one ten-thousandth of the distance between the Earth and the Sun. Perhaps most importantly, the system preserved the total energy of the entire swarm with near-perfect accuracy, proving that it did not artificially create or destroy energy as it calculated the paths. This level of stability is crucial for planetary defense, as it ensures that when we calculate the risk of a collision, the uncertainty in our predictions does not grow uncontrollably over time.

The implications of this work extend beyond just tracking rocks. By being able to accurately reconstruct the paths of millions of objects even when they are hidden, astronomers can build a much more complete and reliable map of our solar system. This is essential for planning future space missions and for ensuring that we can spot a potential threat with enough time to act. The system's ability to handle the complex, shifting gravity of the solar system without losing its mathematical footing suggests a new era in how we model the cosmos. It moves away from methods that approximate the laws of physics and toward a system that embodies them, allowing us to see clearly through the darkness of the observational gaps that have long obscured our view of the universe.

Technical Summary: Hard Symplectic Graph Neural ODEs for Scalable Trajectory Reconstruction

1. Problem Statement

The paper addresses a critical challenge in computational astrodynamics: the continuous-time trajectory reconstruction of large-scale celestial swarms ( $N \ge 10^6$ ) under severe observational constraints. Specifically, the problem arises when observational ephemerides suffer from "blackout" gaps (e.g., solar conjunction or telescope blinding) spanning 25–40% of an orbital period.

Existing approaches face two primary failure modes:

Standard Data-Driven Models: Neural ODEs and RNNs (LSTMs, Transformers) lack inductive physical biases. When propagating trajectories through unobserved gaps, they exhibit catastrophic secular phase-drift, spurious orbital expansion, and violation of conservation laws.
Soft-Regularized Physics-Informed Neural Networks (PINNs): While PINNs attempt to enforce physical laws via soft penalty losses ( $L_{total} = L_{data} + \lambda L_{phys}$ ), empirical evidence suggests this approach fails in long-horizon gap completion. The gradient competition between reconstruction error and conservation residuals leads to non-conservative divergence ( $\nabla \cdot f_\theta \neq 0$ ), causing artificial energy dissipation and phase-space volume erosion that violates Liouville's theorem.

Furthermore, traditional methods often rely on oversimplified assumptions, such as a stationary Sun (ignoring Solar Reflex Motion) and monolithic two-body dynamics (ignoring General Relativity, planetary oblateness, and non-gravitational forces), rendering them unsuitable for high-precision reconstruction of real-world objects like asteroid 99942 Apophis.

2. Methodology: SympG-NODE

The author proposes SympG-NODE (Symplectic Graph-Hamiltonian Neural Ordinary Differential Equations), a structure-preserving deep learning framework designed to guarantee exact mathematical preservation of phase-space symplectic geometry.

2.1 Physical Formulation

Reference Frame: The model operates in the Solar System Barycenter (SSB) inertial frame. Unlike heliocentric models, it treats the Sun, Jupiter, Saturn, and Earth as dynamic nodes satisfying $\sum m_i q_i = 0$ , capturing dynamic Solar Reflex Motion.
Hamiltonian Dynamics: The system is governed by a total specific Hamiltonian $H_{total}$ $H_{t o t a l}$ comprising:
- Kinetic energy $T(p)$ .
- Multi-body Newtonian potential $V_{Nbody}$ .
- Post-Newtonian (1PN) General Relativistic Schwarzschild precession.
- Non-gravitational Solar Radiation Pressure (SRP) forces.
Equations of Motion: Canonical equations are derived via exact automatic differentiation: $\dot{q} = \partial H_\theta / \partial p$ and $\dot{p} = -\partial H_\theta / \partial q + F_{SRP}$ .

2.2 Neural Architecture

Hybrid Hamiltonian Parameterization: The total Hamiltonian is split into an analytic Newtonian backbone and a learned residual perturbation potential $V_\theta$ . This allows the network to focus solely on complex higher-order perturbations (1PN, $J_2$ , secondary perturbations).
Equivariant Graph Neural Network (EGNN): $V_\theta$ is parameterized via an SE(3)-invariant and SO(3)-equivariant GNN. It constructs invariant geometric features (distance, relative velocity, angular momentum) for edges connecting minor bodies to primary planetary nodes.
Adaptive Hierarchical Routing: A key innovation is the Laplace Sphere of Influence (SOI) routing mechanism. The model computes the Hill Sphere radius ( $R_{SOI}$ ) for each major planet and uses a differentiable sigmoid activation to weight message passing. This dynamically routes interactions between global heliocentric orbits and local bound planetary flybys, preventing chaotic divergence in multi-body regions.

2.3 Integration Scheme

To preserve symplectic geometry, the framework employs the Symplectic Implicit Midpoint method. This integrator is a member of the symplectic Runge-Kutta family, ensuring that quadratic invariants are preserved and that energy drift remains strictly bounded ( $O(\Delta t^2)$ ) without secular growth, unlike explicit Runge-Kutta 4 (RK4) schemes.

3. Key Contributions

Hard Symplectic Inductive Bias: Unlike soft-loss PINNs, SympG-NODE enforces hard structural constraints by parameterizing a scalar invariant Hamiltonian, guaranteeing energy conservation ( $|\Delta H|/|H_0| < 10^{-4}$ ) and vanishing phase-space divergence.
Dynamic Barycentric Modeling: The framework explicitly models the dynamic motion of the Sun and major planets, correcting reference frame errors inherent in stationary Sun assumptions.
Hierarchical SOI Graph Routing: The introduction of an adaptive routing mechanism based on planetary Hill Spheres allows the model to seamlessly transition between bound and unbound orbital states.
Unprecedented Scalability: The architecture achieves linear $O(N)$ memory scaling, successfully processing a swarm of 1,000,000 celestial bodies on a standard workstation GPU in under 1.5 seconds per evaluation without VRAM overflow.
Dual Validation: The framework is validated on both authentic NASA JPL Horizons ephemerides (Asteroid 99942 Apophis) and a massive synthetic simulation of 1M bodies.

4. Experimental Results

4.1 Large-Scale Swarm Simulation ( $N=1,000,000$ )

Evaluated on a synthetic swarm comprising Main Belt asteroids, Jupiter Trojans, Near-Earth Asteroids (NEAs), and Centaurs with a 35% observational blackout gap:

Accuracy: SympG-NODE achieved a mean position error of $1.8005 \times 10^{-4}$ AU, outperforming 8 diverse baselines including Soft PINNs, HNN-RK4, SympNet, and PINO-FNO.
Conservation: It maintained an energy drift of $8.42 \times 10^{-5}$ and near-zero phase-space volumetric dissipation ( $3.31 \times 10^{-7}$ ).
Efficiency: Inference latency was 1.48 seconds for the full million-body batch.
Comparison: Soft PINNs exhibited energy drifts exceeding 67%, and non-symplectic models (Vanilla NODE, Bi-LSTM) showed catastrophic divergence.

4.2 Real-World Validation (99942 Apophis)

Validated on authentic NASA JPL Horizons data for the potentially hazardous asteroid 99942 Apophis, covering a 60-day blackout gap during its 2029 close approach:

Performance: SympG-NODE maintained sub-millimeter position accuracy ( $\|\Delta q\| \le 1.8 \times 10^{-4}$ AU) and strict energy preservation ( $|\Delta H|/|H_0| = 8.42 \times 10^{-5}$ ).
Baseline Failure: Non-symplectic models suffered severe unphysical phase lag and energy divergence (>500%), while even symplectic baselines like HNN-RK4 accumulated numerical dissipation due to explicit integration.

5. Significance and Implications

The paper claims that SympG-NODE provides a mathematically rigorous foundation for neural orbital hazard mitigation, particularly for planetary defense. Its significance lies in:

Liouville's Theorem Preservation: By maintaining phase-space volume, the model ensures that orbital uncertainty ellipsoids (critical for impact probability calculations) are not artificially contracted or diluted, a flaw present in non-symplectic deep learning models.
Velocity Recovery: The Hamiltonian formulation allows for the direct derivation of instantaneous 3D velocity vectors via exact automatic differentiation, avoiding the noise amplification associated with finite differencing of astrometric data. This is crucial for accurate kinetic energy estimation in impact scenarios.
Scalability: The ability to process one million bodies in real-time on standard hardware enables the monitoring of vast celestial swarms without requiring supercomputing infrastructure.

The author concludes that this framework overcomes the limitations of soft-loss regularization and simplified dynamics, offering a robust solution for trajectory reconstruction in perturbed relativistic N-body systems under realistic observational constraints. Future work is noted to extend the framework to non-conservative Yarkovsky effects and higher-order post-Newtonian expansions.