Source-linked AI summary
Continuous Learning of Gravity Field Irregularities Around Small Bodies via Neural Hamiltonian ODEs
Giacomo Acciarini, Dario Izzo
TL;DR
Small-body gravity must be inferred indirectly from noisy tracking, and earlier neural models relied on field labels rather than mission tracking arcs. This paper estimates a neural Hamiltonian from tracking with exact variational gradients; its reported model remains useful inside the Brillouin sphere, where harmonics errors can grow with degree.
Problem
Gravity is observed through noisy position and velocity arcs rather than directly, while prior neural gravity models relied on field labels, leaving tracking-only estimation on flyable arcs at realistic noise levels unestablished.
Source: §1 · pp. 1–2Method
A feed-forward network represents the unknown part of the Hamiltonian, and exact variational equations provide training gradients while harmonics are estimated through the same workflow.
Source: §1 · p. 2Results
The neural model is reported valid inside and outside the Brillouin sphere and better than harmonics for descent; in the 67P deep region, relative field error was 0.29 in the deepest bin and 0.51 at worst, versus degree 4 at 0.65, degree 8 at 1.34, and degree 12 at 45.
Source: §1 · p. 2, §3.1 · p. 8Takeaways & Limitations
The method serves descent and proximity operations from tracking before a shape model exists, then incorporates the imaged shape as known physics when available.
Source: §5 · p. 14High-degree harmonics are weakly observable from orbit, leaving their coefficients nearly unconstrained and able to absorb noise that is amplified near the surface.
Source: §2.3 · p. 5
Abstract
from arXiv · showhide
We propose to learn the unknown dynamics in the proximity of a small body directly from tracking data, representing them as a feed-forward neural network embedded in the system Hamiltonian. The equations of motion form a Neural Hamiltonian Ordinary Differential Equation, whose variational equations provide exact training gradients: estimation uses position and velocity arcs at realistic noise levels, without acceleration or potential labels, and a continual learning approach warm-starts the network as new data are acquired. The known part of the Hamiltonian carries whatever is available, from the central term and spin state to the constant-density model of the imaged shape. We assess the method against a normalized spherical harmonics expansion estimated from identical arcs through the same machinery, on scenarios built on the shapes of Itokawa, 67P, Bennu and Eros. The network remains usable inside the Brillouin sphere: it plans ballistic descents at Itokawa to \SI{4.6}{m} median touchdown error from tracking alone, against 5.1--\SI{48.9}{m} for harmonics of degree 4--12, and to \SI{0.9}{m} with the imaged shape as prior, a configuration that also recovers localised density anomalies invisible to any harmonics degree. The two representations are complementary, and we discuss their combined use across the phases of a small-body mission.
1 Introduction
Small-body gravity must be inferred through noisy trajectory tracking, while spherical harmonics have a structural convergence limit inside the Brillouin sphere. The introduction proposes estimating a neural Hamiltonian from the same dynamics-based workflow, with available shape physics and continual learning as data accumulate.
1 Introduction
- Gravity is a science product and a force-model input for proximity guidance, yet spacecraft infer it through trajectories because they do not directly measure gravity.Operational estimation must use noisy observed quantities such as position and velocity, at their realistic noise levels.Source: §1 · p. 1
- Spherical harmonics are a tracking-based baseline, but their expansion converges only outside the Brillouin sphere, and proximity operations therefore often switch to shape-derived polyhedral or mascon models.Those proximity models are not estimated from tracking, and assumed interior density makes their errors largest for signatures of internal irregularities.Source: §1 · pp. 1–2
- Earlier neural gravity models demonstrated accuracy on irregular bodies but relied on acceleration or potential labels, leaving tracking-only estimation on flyable arcs at realistic noise levels unestablished.The paper also asks whether such models can assimilate tracking as a mission progresses and how they compare with a classical pipeline fitted to identical data.Source: §1 · p. 2
- The unknown Hamiltonian component is represented by a feed-forward network, while available physics enters the known component; exact variational equations supply training gradients, and harmonics use the same estimation workflow.The known component can include the central term and spin state, later the constant-density imaged shape, and measured non-gravitational forces throughout.Source: §1 · p. 2
- From tracking alone, the neural Hamiltonian is reported valid inside and outside the Brillouin sphere and better than harmonics for descent; with a shape prior, it recovers localized structures inaccessible to any harmonics degree.The contributions also include joint estimation of non-gravitational Hamiltonian forces, greater robustness to force-model errors, and continual learning as data arrive.Source: §1 · pp. 2–3
2 Methodology
The method estimates a small body's unknown Hamiltonian term from trajectory arcs using a neural potential correction and exact variational gradients. It warm-starts fits as observations accumulate and can select informative future orbit families.
2.1 Dynamics
- The learned correction maps body-fixed position to a scalar potential, while the known Hamiltonian includes available terms and a radial envelope makes the correction fade with altitude.The envelope imposes the physical 1/r^3 falloff; the correction is not constrained to be harmonic.Source: §2.1 · p. 3
2.2 Backpropagating through the Flow
- Batch least squares fits noisy position–velocity arcs anchored at their first observed states, integrating state sensitivities jointly to compute exact gradients through variational equations.Source: §2.2 · pp. 4–5
2.3 Solvers
- The harmonics coefficients are fit with damped Gauss–Newton, while neural parameters use mini-batched Adam; both solvers use the same variational-equation Jacobians.Source: §2.3 · p. 5
- Because high-degree harmonics are weakly observable from orbit and can absorb noise that amplifies near the surface, the baseline uses a Kaula-type degree penalty where beneficial.For Section 4.3, the penalty weight is scanned and the best truth-based evaluation value retained.Source: §2.3 · p. 5
2.4 Continual Learning Approach
- As data arrive in stages, both representations are warm-started on accumulated observations, while candidate orbit families are greedily ranked by predicted D-optimal information gain without measurements or truth.Source: §2.4 · p. 6
- At fifteen orbits, D-optimal selection reaches 0.14, 1.6× below the best random order at equal budget and below every unselected campaign even with eighteen.Source: §2.4 · p. 6
3 Results From Tracking Alone
The tracking-only experiments fit gravity models without a shape model, using position–velocity arcs and evaluating the inferred field against simulation truth. Across 67P, Itokawa, and Bennu scenarios, the network remains useful in deep-region extrapolation and descent planning, while the results also test joint estimation with solar radiation pressure.
3 Results From Tracking Alone
- The fitted models use position–velocity arcs alone, without field quantities; relative field error is evaluated against simulation truth over shells and surface-following altitude sets.At a fixed radius, shells provide a comparison against the no-correction baseline; altitude sets test terminal-descent regimes.Source: §3 · p. 7
3.1 Inside the Brillouin Sphere: the 67P Scenario
- On 67P, the network’s relative field error stayed between 0.29 and 0.51 in the deep region, while degree-8 and degree-12 harmonics reached 1.34 and 45 despite fitting identical arcs.All training samples were at or above 0.96 R_B, so the deep-region evaluation was extrapolation.Source: §3.1 · p. 8, Figure 4 · p. 8
- The 67P equatorial-plane comparison shows harmonics errors exceeding the no-correction level in surface-adjacent patches, while the network’s median error on those patches was 0.27.The network exceeded unit error only in a thin band within a few hundred metres of terrain.Source: §3.1 · p. 8, Figure 5 · p. 9
- After nine 67P descent arcs were assimilated, the network’s deepest-bin error fell from 0.29 to 0.15; degree 8 remained at 1.5 and degree 12 at 7.6.The same descent data were added by warm-refitting.Source: §3.1 · p. 8, Figure 6 · p. 9
3.2 Ballistic Descent Planning
- 4.6 m median and 21.2 m worst-case miss: the network landed all 30 Itokawa descents, while degree-4, -8, and -12 harmonics had 5.1, 15.9, and 48.9 m median miss, respectively.Nine descents never reached the surface under degree 12.Source: §3.2 · p. 9, Figure 7 · p. 10
- With 672 Itokawa arcs, the network’s median miss improved from 4.6 to 2.7 m and all descents landed, while degree-8 error at 30 m altitude remained 3.7.The extended campaign used 48 orbits and 672 arcs.Source: §3.2 · pp. 9–10
3.3 Joint estimation of gravity and solar radiation pressure
- For the Bennu particle swarm, joint SRP estimation recovered per-particle area-to-mass ratio with 0.6% median error for degree-8 harmonics and 3.5% for the network.Both fits estimated the 28 particles’ SRP magnitudes jointly with gravity.Source: §3.3 · p. 11, Figure 8 · p. 11
- When SRP was omitted from Bennu fits, harmonics field error at 30 m altitude reached 3.5, while the network error increased from 0.33 to 0.48.The learned potential’s r^-3 decay made the network more resistant to absorbing the unmodelled force.Source: §3.3 · p. 11
4 Results with the Imaged Shape as a Prior
With the imaged shape in the known Hamiltonian, the network estimates only the interior residual from tracking. The residual architecture improves descent planning and recovers localized anomalies that harmonics represent poorly.
4 Results with the Imaged Shape as a Prior
- The shape model is fixed in the known Hamiltonian, leaving the network to estimate the interior residual from tracking without refitting the masses.Because truth and prior share total mass and are centered on the center of mass in these experiments, the residual has no monopole or dipole; the estimator is unchanged.Source: §4 · pp. 11–12
4.1 The Interior Residual and the Choice of Architecture
- The residual network improved on the shape prior at every altitude and on its from-scratch fit by a factor of 4.4.At 1.05 R_B, the from-scratch fit had 0.489 relative field error versus 0.160 for the constant-density shape model; Figure 9 compares the architectures.Source: §4.1 · p. 12, Figure 9 · p. 12
4.2 Descent planning with the residual model
- 0.9 m median and 6.0 m worst-case miss, with no failures, made the network residual the campaign’s best descent model and 3.4× better than the shape model alone.The result was about twice as good as the next-best campaign result.Source: §4.2 · p. 12
4.3 Localized density anomalies
- 0.17–0.19 relative field error on every shell was 3.5× better than the best harmonics option across degrees and regularizations; Figure 10 shows localized anomaly recovery.The shape prior alone had 0.76 error, and the network recovery is concentrated where the anomalies are located.Source: §4.3 · p. 13, Figure 10 · p. 13
5 Conclusions
The paper positions neural Hamiltonians and spherical harmonics as complementary gravity representations for different small-body mission phases. The neural approach supports proximity operations and continual updates, while harmonics remain more accurate for smooth exterior gravity observed from altitude.
5 Conclusions
- The method estimates the unknown dynamics as a neural Hamiltonian from tracking arcs using the exact variational equations of a Taylor integrator.Its performance was assessed against spherical harmonics estimated from identical data through identical numerics.Source: §5 · p. 13
- The representations fail in disjoint regimes: harmonics are more accurate for smooth exterior gravity, while the neural method serves proximity operations and recovers buried anomalies.The neural model also degrades mildly rather than catastrophically under force-model mis-modelling.Source: §5 · pp. 13–14
- Warm refits assimilate each new tracking batch, and the solvers’ information matrix can score candidate orbits before flight.Source: §5 · p. 14