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Constrained Diffusion for Data-Scarce Orbital Monte Carlo in Constellation Tasking

Omar Ramadan, Sam Siavoshian, Amir Kashif Saeed, Benjamin A. Johnson, Amin M. E. -A. Diab, Benjamin M. Rodriguez

arXiv:2610.09323v1astro-ph.IMastro-ph.EPcs.AI

TL;DR

Monte Carlo campaign credibility depends on the supplied trajectory population, yet scarce references make replay repetitive and independent element jitter can violate physical constraints. The paper learns a force-conditioned orbital prior and propagates samples with Basilisk. Diffusion matches strong simple baselines on held-out statistics within support, but does not solve support shift or replace classical astrodynamics.

  • Problem

    Monte Carlo custody results depend on the supplied trajectory population, but scarce references make replay repetitive and independent orbital-element jitter can violate physical constraints.

    Source: §1 · p. 1

  • Method

    A force-conditioned diffusion model learns an orbital prior, preserves perigee geometry by deriving semimajor axis from perigee altitude, and delegates temporal evolution to Basilisk.

    Source: §1 · p. 2

  • Results

    Diffusion ties jittered bootstrap and per-tier Gaussian mixtures on held-out trajectory MMD while sampling farther from training priors; all generators fail prescribed support shift, and diffusion is indistinguishable from replay and bootstrap on mean custody.

    Source: §6 · p. 13

  • Takeaways & Limitations

    Constrained diffusion is a complementary method for locally supported Monte Carlo augmentation, while trajectory-population construction remains a consequential source of uncertainty in constellation analysis.

    Source: §6 · p. 14

    The synthetic 800-trajectory corpus is not representative of an operational catalog, and reported intervals do not measure uncertainty over an operational population.

    Source: Limitations and Threats to Validity · p. 13

Abstract

from arXiv · show

Constellation Monte Carlo results depend on the orbital population used to evaluate a tasking policy. With scarce reference trajectories, replay limits geometric diversity, while independent orbital-element jitter can violate physical constraints. We study constrained diffusion for orbital-population augmentation. A force-conditioned diffusion model learns a 13-dimensional orbital prior, recovering semimajor axis from perigee altitude and eccentricity; Basilisk propagates each sample under one of five force-model tiers. Using 800 reference trajectories, we compare diffusion with jittered bootstrap, per-tier Gaussian mixtures, and a conditional variational autoencoder, and evaluate distributional fidelity, support shift, classical astrodynamics diagnostics, and 4,000 paired GoDSAT-compatible campaigns. The largest diffusion model achieves held-out trajectory MMD of 0.0171 +/- 0.0239, similar to bootstrap (0.0170) and the mixture (0.0198), but samples farther from training priors (median nearest-training distance 2.7 versus 0.10 standardized units). All generators fail to match the shifted-blind population (classifier AUC 0.991-0.998). Endpoint-conditioned samples satisfy Lambert boundaries but have greater interior error than the matched Lambert reference (29.2 versus 5.7 km mean RMSE). Residual diffusion improves selected sparse forecasts and catalog-mode recall but does not outperform classical estimators on custody ranking. In a fixed 16-satellite configuration, diffusion yields similar mean custody to replay and bootstrap, while the force-tier mixture shifts custody by about 5.5 percentage points. Constrained diffusion supports local, in-support augmentation but cannot replace orbital dynamics or serve as an operational posterior. Orbital-population construction is a consequential source of uncertainty in constellation analysis.

1. INTRODUCTION

Monte Carlo custody results depend on the trajectory population, but scarce references make replay repetitive and independent element jitter physically unsafe. This paper tests constrained diffusion for orbital-population augmentation while separating executability, distributional fidelity, and decision utility.

Motivation

  • A Monte Carlo custody result can reflect the supplied orbit set rather than the tasking logic when that set is small, repetitive, or geometrically inconsistent.Custody transfer, tracking continuity, and handoff timing depend on interactions between the policy, orbits, sensor geometries, and observation opportunities.Source: §1 · p. 1
  • Scarce reference sets make replay geometrically repetitive, while independent orbital-element jitter can violate coupled constraints and create Earth-intersecting perigees.The relevant goal is expanding a physically admissible population within declared simulation support, not merely generating more orbits.Source: §1 · p. 1

Research question

  • The paper tests whether force-conditioned diffusion can expand a data-scarce orbital population and identifies where augmentation becomes unreliable relative to classical astrodynamics.The model learns an orbital prior and delegates temporal evolution to Basilisk; semimajor axis is recovered from perigee altitude to avoid Earth-intersecting combinations.Source: §1 · p. 2

Contributions

  • The study contributes a constrained orbital prior, staged validation against statistical and astrodynamics baselines, and a GoDSAT-compatible campaign testing population effects on custody.It treats downstream custody as a property of the induced population, not a score to maximize.Source: §1 · p. 2

2. RELATED WORK

Prior tasking and custody studies generally treat trajectory populations as inputs, while operational sampling methods can be brittle when references are scarce. This paper focuses on constrained in-support augmentation and evaluates it against simple generators and downstream custody.

Tasking and custody evaluation

  • Tasking and custody studies use trajectory populations as inputs and do not examine whether Monte Carlo orbits themselves introduce uncertainty into policy scoring.The cited tasking literature includes space-based and ground-based schedulers, with custody, covariance, and unique-object count among typical figures of merit,,,.Source: §2 · p. 2

Operational sampling and classical references

  • Replay can under-cover handoff geometry and element-wise jitter can break coupled geometry, while Basilisk can propagate sampled initial conditions under specified force models.Lambert transfers, filtering and smoothing, and batch orbit determination remain physical references for learned corrections,.Source: §2 · p. 2

Generative trajectory models

  • This paper generates a constrained orbital prior for simulator rollout rather than a trajectory sequence, testing in-support augmentation against per-tier Gaussian mixtures and jittered bootstrap.The stated gap is measured using distributional tests and downstream custody, not a claim that diffusion is the preferred orbital-element density model,.Source: §2 · pp. 2–3

3. METHODOLOGY

The study frames augmentation as generating many synthetic trajectories from a sparse reference set within declared simulation support. It separates geometric, support, dynamic, and observation or decision constraints because satisfying one does not establish the others.

Problem Formulation and Monte Carlo Use Case

  • Augmentation is useful only when synthetic trajectories are executable, faithful to relevant held-out cases within support, and informative for downstream sensitivity analysis.These criteria concern local augmentation quality and do not imply recovery of an unknown operational population.Source: Problem Formulation and Monte Carlo… · p. 3
  • The campaign compares four frozen populations—seed replay, bootstrap, force-tier Gaussian mixtures repropagated through Basilisk, and constrained diffusion—using paired campaign seeds.Each population contains 1,000 trajectories, and shared GoDSAT stochastic seeds isolate population-construction contrasts under common campaign randomness.Source: Problem Formulation and Monte Carlo… · p. 3

Trajectory Constraint Contract

  • The constraint contract combines valid orbital geometry, declared training support, Basilisk propagation, and observation or decision requirements.Observation and decision constraints are separate because an executable trajectory may still be difficult to track or maintain in custody.Source: Trajectory Constraint Contract · p. 3
  • Basilisk execution establishes consistency with the declared rollout model, not simulator correctness, operational-population agreement, or validity under regime shift.Physical admissibility, distributional fidelity, extrapolation behavior, and downstream decision utility are measured separately.Source: Trajectory Constraint Contract · p. 3

Classical Astrodynamics and Lambert Reference

Classical propagation and estimation provide references for evaluating learned corrections, while Lambert solutions serve as a targeted endpoint-conditioned consistency test. The benchmark varies force tiers and support, and the learned prior encodes orbital parameters to preserve valid geometry before simulator propagation.

Classical Astrodynamics and Lambert Reference

  • Classical propagation, filtering and smoothing, batch orbit determination, and multistart estimation test whether learned residuals add value after dominant orbital structure is removed.Lambert’s problem is a boundary-value reference, not a baseline for unconditional population generation; its branch selection here uses the reference trajectory and is not operational.Source: Classical Astrodynamics and Lambert… · p. 3
  • Endpoint-conditioned diffusion imposes exact endpoint positions, but its interior errors measure consistency with a matched Lambert arc rather than universal physical accuracy.The experiment treats boundary satisfaction as necessary for the conditioning task, not sufficient evidence of an interior Lambert-consistent trajectory.Source: Classical Astrodynamics and Lambert… · p. 3

Basilisk Benchmark

  • The Basilisk benchmark has 800 trajectories across five force tiers, with 512 training, 96 validation, 96 test, and 96 shifted-blind cases.The blind split changes perigee altitude, eccentricity, inclination, and duration support while retaining the declared ranges for orientation angles and spacecraft properties.Source: Basilisk Benchmark · p. 3, Figure 1 · p. 4

Constrained Orbital-Prior Representation

  • The prior encodes orbital parameters with angular sine/cosine pairs and derives semimajor axis from perigee altitude to prevent Earth-intersecting perigee geometry.Channels are standardized on training data and decoded within per-tier training bounds; epoch is excluded from learned-distribution metrics to avoid index leakage.Source: Constrained Orbital-Prior Representation · p. 4

Conditional Diffusion and Generative Baselines

The method corrupts standardized orbital priors with a DDPM and conditions denoising on diffusion step and force tier. It is evaluated against a conditional VAE, per-tier Gaussian mixtures, and jittered bootstrap.

Conditional Diffusion and Generative Baselines

  • The DDPM corrupts standardized priors, while the denoiser uses diffusion-step and force-tier embeddings and minimizes noise-prediction mean squared error.The passages specify the corruption and conditioning but do not give further denoiser architecture details.Source: Conditional Diffusion and Generative… · p. 4
  • The diffusion models vary hidden width across 64, 128, and 256, with 20,525, 40,749, and 105,773 parameters, respectively.Training uses 1,200 optimizer steps, batch size 128, AdamW, a cosine schedule, and 64 diffusion steps.Source: Conditional Diffusion and Generative… · p. 4
  • The comparison baselines are a 40,989-parameter conditional VAE, per-tier full-covariance mixtures with at most four components, and jittered bootstrap with standardized noise σ = 0.03.These simple statistical baselines are motivated by the prior's low dimensionality and smoothness.Source: Conditional Diffusion and Generative… · p. 4

Distributional Metrics and Statistical Units

The section defines distributional comparisons and their inferential unit, then specifies sparse-observation estimation experiments, calibration, and evaluation measures. The diagnostics use held-out trajectories and classical propagation references.

Distributional Metrics and Statistical Units

  • Distributional evaluation reports unbiased RBF MMD, permutation p values, fivefold random-forest C2ST AUC, marginal Wasserstein distance, and nearest-training-prior distance.MMD and C2ST use equal sample counts selected without replacement for each generated set and reference tier.Source: Distributional Metrics and Statistical… · p. 5
  • The primary inferential unit is a seed–force-tier block (n = 25), compared with paired Wilcoxon tests; exploratory per-tier diagnostics receive no multiplicity correction.Prespecified scaling contrasts compare 64 versus 512 trajectories on test and blind splits.Source: Distributional Metrics and Statistical… · p. 5

Downstream Diagnostic I: Sparse Observations

  • Sparse-observation tests use 64-step ECI sequences, with smoothing masks retaining 10%, 25%, or 50% of states and forecast masks retaining the first 25% or 50%.Position noise has standard deviation 100 or 1000 m; metrics are calculated only at unobserved smoothing states or after the final forecast observation.Source: Downstream Diagnostic I: Sparse… · p. 5
  • Learned temporal U-Nets and a Transformer correct RTS or two-body trajectories using noisy positions, masks, time, duration, noise level, force tier, and a complete reference trajectory.Classical references are linear interpolation or extrapolation, a constant-velocity RTS smoother, and central-gravity Runge–Kutta propagation initialized from the RTS state.Source: Downstream Diagnostic I: Sparse… · p. 5
  • Split-conformal calibration uses 114 calibration trajectories, and evaluation includes ensemble-mean position and velocity RMSE, energy score, coverage, sharpness, consistency, and central-gravity residual.The local custody gate requires position error below 100 km and radial 90% spread below 200 km, and is a controlled surrogate rather than an operational custody definition.Source: Downstream Diagnostic I: Sparse… · p. 5

Downstream Diagnostic II: Ambiguous Tracking

The ambiguous-tracking diagnostic uses nonlinear ground-station observations and declared catalog alternatives. Physics-residual diffusion is compared with classical estimators using calibrated recovery and custody-ranking measures.

Downstream Diagnostic II: Ambiguous Tracking

  • Eight rotating spherical-Earth ground stations provide nonlinear measurements, with range-only and angles-only regimes observing 10% of epochs and a mixed regime observing range, range rate, azimuth, and elevation at 25% density.Observations are confined to the first half arc, leaving the terminal custody handoff unobserved.Source: Downstream Diagnostic II: Ambiguous… · p. 5
  • Each case receives two catalog hypotheses, including a noisy near-object state and a regime-specific alternate, with a 10% robust association prior in explicitly ambiguous regimes.The alternatives are generated before estimation and are available identically to all methods.Source: Downstream Diagnostic II: Ambiguous… · p. 5

Classical Estimators, Physics-Residual Diffusion, and Calibration

  • The declared catalog ambiguity does not establish that the learned ensemble discovers all modes of an unknown operational posterior.Classical comparators are a central-gravity EKF with RTS smoothing, sigma-point UKF smoother, regularized nonlinear batch orbit determination, and a two-start batch candidate ensemble.Source: Downstream Diagnostic II: Ambiguous… · p. 5, Classical Estimators, Physics-Residual… · p. 5
  • The physics-residual U-Net learns six-channel corrections around each refined component, and the test energy score favored the U-Net over the Transformer, 0.01011 versus 0.01237.The Transformer was reported as a negative architecture ablation; residual shrinkage and a finite-sample calibration multiplier are selected using calibration trajectories.Source: Classical Estimators, Physics-Residual… · pp. 5–6
  • Recovery measures include energy score, several RMSE variants, BIC support, and catalog-mode recall; custody ranking is assessed with selective-risk AUC, Brier score, and acceptance measures.The fixed custody label is modal terminal position error at most 100 km, and 100 paired inferential units per split are formed by averaging six strata within each seed.Source: Classical Estimators, Physics-Residual… · p. 6

GoDSAT-Compatible Monte Carlo Protocol

The end-to-end experiment compares four frozen orbital populations in paired GoDSAT-compatible campaigns. It uses a documented Octave compatibility port, so results do not certify native MATLAB/Mapping Toolbox parity.

GoDSAT-Compatible Monte Carlo Protocol

  • Four frozen 1,000-trajectory populations—seed replay, bootstrap, force-tier GMM samples repropagated with Basilisk, and constrained diffusion—are evaluated through the campaign.Each GoDSAT campaign index uses the same stochastic seed across methods, making campaign index the pairing unit.Source: GoDSAT-Compatible Monte Carlo Protocol · p. 6
  • The common campaign uses 16 satellites and a predeclared failure threshold of custody fraction below 0.80.Other fixed settings include a 2-s simulation step, 20° fields of view, and a 90 × 180 ClaimMap.Source: GoDSAT-Compatible Monte Carlo Protocol · p. 6
  • The documented GNU Octave compatibility port preserves specified GoDSAT logic while adding external ECI-trajectory input and JSON metric export, but does not certify native MATLAB/Mapping Toolbox parity.Mean intervals use 10,000 nonparametric bootstrap resamples, with paired contrasts using common campaign indices.Source: GoDSAT-Compatible Monte Carlo Protocol · p. 6

4. RESULTS

Constrained diffusion matches strong simple baselines on held-out interpolation and produces samples farther from individual training priors, but it fails under prescribed support shift. Its forecasting and multimodal-recall gains are selective, custody ranking does not improve, and population choice changes estimated custody in the GoDSAT-compatible campaign.

Data Scaling and Support Shift

  • All six generators fail shifted-blind equality tests; DDPM-256 median nearest-training distance is approximately 2.7 standardized units versus 0.10 for jittered bootstrap.Increasing training data improves held-out interpolation substantially but does not produce reliable extrapolation; degree-8 gravity with drag is hardest across methods, with MMD 1.02–1.09 versus 0.51–0.61 for J2.Source: Data Scaling and Support Shift · p. 6

Held-Out Interpolation

Constraint Representation and Lambert Consistency

  • The final codec derives semimajor axis from perigee altitude, and all 28,800 generated samples propagate successfully; exact Lambert endpoints still leave diffusion interior position RMSE 23.459 km higher than reference.The Lambert comparison is a targeted consistency test, not an absolute physical-accuracy ranking, because some reference tiers include non-Keplerian perturbations.Source: Constraint Representation and Lambert… · pp. 6–7, Table 3 · p. 8

Downstream Diagnostic: Sparse Observations

  • On held-out forecasting, the two-body-residual U-Net reduces position RMSE from 143.2 to 73.1 km with 25% observations and from 88.5 to 54.1 km with 50% observations.RTS or two-body propagation remains best in smoothing; blind forecast reductions are not significant at the regime level, and the overall paired advantage is not significant because dense-smoothing blocks are lost.Source: Downstream Diagnostic: Sparse… · pp. 7–8, Table 4 · p. 8

Downstream Diagnostic: Ambiguous Tracking

  • Mode recall exceeds multistart by 0.206 on test and 0.238 on blind, but test false-accept rate is 0.0455 for diffusion versus 0.0152 for multistart.Diffusion is worse in mean energy than batch MAP and multistart; on blind data, its false-accept rate is 0.0424, statistically tied with batch MAP at 0.0428.Source: Downstream Diagnostic: Ambiguous… · p. 8, Figure 8 · p. 11

Fresh-Split Capacity–Budget Confirmation

  • On the fresh split, width 640 is worse than width 160 by 0.254 km on test and 0.424 km on blind, and capacity or budget contrasts do not materially improve decision-level performance.Longer training modestly improves best-of-set recovery, but test false-accept remains 0.0545–0.0556 across diffusion cells versus 0.0212 for multistart.Source: Fresh-Split Capacity–Budget Confirmation · pp. 10–11

GoDSAT Population-Sensitivity Evaluation

  • In the paired campaign, constrained diffusion is statistically indistinguishable from seed replay and bootstrap on mean custody, while the force-tier GMM yields custody higher by approximately 5.46 percentage points.These are conditional frequencies under the synthetic stress campaign; the population difference is a modeling choice, not evidence that the GMM is a better generator.Source: GoDSAT Population-Sensitivity Evaluation · p. 11

5. DISCUSSION

The discussion finds a narrow role for constrained diffusion: it can diversify locally supported orbital populations, but does not replace classical astrodynamics or establish operational validity. In the GoDSAT-compatible campaign, population choice changes estimated custody, while synthetic-data, physical-validation, and runtime boundaries constrain interpretation.

What the Results Establish

  • Within observed support, DDPM-256 matches strong low-dimensional baselines in trajectory fidelity while sampling farther from individual training priors; all generators fail under prescribed support shift.The discussion also reports selected forecasting and catalog-mode benefits, but no superiority in dense smoothing or custody ranking.Source: What the Results Establish · p. 11
  • Diffusion, replay, and bootstrap yield similar mean custody, while the force-tier GMM shifts the estimate upward by approximately 5.46 percentage points; this reflects population differences, not sampler quality.The comparison uses the same campaign configuration, and methods induce different synthetic evaluation populations.Source: What the Results Establish · p. 11, Table 7 · p. 12

Why Diffusion When Simple Baselines Are Strong?

  • The 13-dimensional prior makes simple statistical generators strong competitors; diffusion’s value is comparable fidelity with greater distance from training priors and reuse in residual forecasting and multimodal candidate representation.The GMM has the lowest parameter MMD, and jittered bootstrap matches the strongest DDPM on held-out trajectory MMD.Source: Why Diffusion When Simple Baselines… · p. 12

Constellation-Level Use

  • Diffusion is intended to expand evaluation conditions within declared training support, not replace GoDSAT’s tasking logic; synthetic-population frequencies are not operational occurrence probabilities without external calibration.A higher custody score may reflect an easier population, so generated-population quality and downstream task difficulty are distinct.Source: Constellation-Level Use · p. 12

Scope of Physical Validity

  • Basilisk repropagation establishes consistency with the selected simulator, while endpoint equality does not establish Lambert-consistent interiors or independent physical validity.Stronger validation would require a separate high-fidelity engine and comparison with external ephemerides or controlled observations.Source: Scope of Physical Validity · pp. 12–13

Limitations and Threats to Validity

  • The 800-trajectory synthetic corpus and simplified observation assumptions do not represent an operational catalog; repeated seeds quantify conditional benchmark variation, not uncertainty over operational populations.Basilisk generates both reference data and rollouts, and the Lambert study is a controlled two-body benchmark rather than an operational transfer-selection procedure.Source: Limitations and Threats to Validity · p. 13
  • The campaign uses an Octave compatibility port without empirical certification of native-runtime parity; paired seeds do not physically match generated populations, and 1,000 runs do not resolve rare ClaimMap conflicts.The local surrogate and synthetic campaign are not mission or operational certification.Source: Limitations and Threats to Validity · p. 13

Follow-On Work

  • Follow-on work calls for broader priors with explicit coverage targets, comparisons with alternative generative models, and calibration against independent populations, observations, dynamics, and native-runtime parity.The present low-dimensional setting may favor explicit-density approaches, making comparison with the strong GMM and bootstrap especially relevant.Source: Follow-On Work · p. 13

6. CONCLUSION

The conclusion presents constrained diffusion as a locally supported augmentation mechanism, not a replacement for orbital dynamics. It reports similar custody for diffusion, replay, and bootstrap, but a shifted estimate for the force-tier mixture, underscoring population construction as a source of uncertainty.

6. CONCLUSION

  • Within observed support, diffusion matches strong baselines in held-out trajectory statistics while sampling farther from training priors; all generators fail a prescribed support shift.Endpoint-conditioned generation meets Lambert boundaries without recovering the interior transfer, and learned residuals do not outperform classical estimators on dense smoothing or custody ranking.Source: §6 · p. 13
  • Across 4,000 GoDSAT-compatible campaigns, diffusion is indistinguishable from seed replay and bootstrap on mean custody, while the force-tier Gaussian mixture shifts the estimate by approximately 5.5 percentage points.The methods induce different synthetic evaluation populations, so a higher custody fraction may simply indicate an easier sampled mix.Source: §6 · pp. 13–14
  • Constrained diffusion is a complementary mechanism for locally supported Monte Carlo augmentation, not an operational posterior or substitute for independently validated astrodynamics.Reliable use requires an externally justified target population and explicit handling of support shift.Source: §6 · p. 14

BIOGRAPHY[

The biographies describe contributors whose work spans data science, machine learning, and engineering. Their stated backgrounds include trajectory modeling, experimental-pipeline development, and research and teaching across AI and analytics.

BIOGRAPHY[

  • Omar Ramadan researches physically grounded AI and has developed trajectory models, Monte Carlo uncertainty methods, and GPU-accelerated learning pipelines.His interests also include computer vision, robotics, and autonomous systems.Source: BIOGRAPHY[ · p. 15
  • Sam Siavoshian contributed to the study’s conceptualization, dataset and experimental-pipeline architecture, and diffusion-model training.His broader interests include persistent memory and continuous computation in AI systems.Source: BIOGRAPHY[ · p. 15
  • Amir K. Saeed’s work spans engineering, machine learning, and data science; Benjamin A. Johnson’s recent research concerns data fusion and reinforcement learning for satellite tasking.Saeed teaches algorithms and data patterns at Johns Hopkins, while Johnson previously served as a U.S. Army Signal Officer.Source: BIOGRAPHY[ · pp. 15–16
  • Amin M. E.-A. Diab researches generative and agentic AI, while Benjamin M. Rodriguez leads multidisciplinary work on sensing, decision-making, and modeling and simulation.Diab’s work spans several application domains; Rodriguez also advances graduate education in data science and analytics.Source: BIOGRAPHY[ · p. 16
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