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.
Figures & tables
Fig. 1: Coverage of the 800-trajectory Basilisk benchmark. The blind split changes orbital and duration support; orientation angles and spacecraft properties retain their declared ranges.
Item
Value
Reference train / test / blind
512 / 96 / 96
Force tiers
5
Training seeds
5
Primary generators
6
DDPM hidden widths
64 / 128 / 256
DDPM training sizes
64 / 128 / 256 / 512
TABLE I: Primary Experimental Matrix
Held-Out Test
Shifted Blind
Generator
Param. MMD
Traj. MMD
C2ST AUC
Param. MMD
Traj. MMD
C2ST AUC
DDPM-64
0.0320
0.0627
0.621
0.5071
0.7118
0.991
DDPM-128
0.0128
0.0275
0.618
0.5063
0.7400
0.995
DDPM-256
0.0122
0.0171
0.641
0.5125
0.7305
0.997
Conditional VAE
0.0754
0.1187
0.757
0.5799
0.8162
0.998
Tier GMM
0.0060
0.0198
0.610
0.5112
0.7448
0.993
TABLE II: Primary Generator Results (Mean Over 25 Seed–Tier Blocks)
Fig. 2: Trajectory MMD across 25 seed–tier blocks per model and split. Points are block values; markers and bars show mean and one standard deviation. Neural complexity does not dominate the simple baselines on held-out interpolation, and all methods degrade sharply under shift.
Fig. 3: Sample-efficiency sweep for the 256-wide DDPM. More training data improve both regimes, but the large blind gap remains. Bars show standard deviation over 25 seed–tier blocks.
Fig. 4: Shifted-blind trajectory MMD by force tier and generator. Degree-8 gravity with drag is the hardest tier for every model. No cell passes its permutation equality test.
TABLE IV: Sparse-Observation Position RMSE and Calibrated Coverage (Means Over Five Seeds and Two Noise Levels)
Fig. 5: Position RMSE by observation regime. Points and bars show means and one standard deviation over five seeds and two noise levels. Physics-residual diffusion is strongest for forecasting, while classical estimators dominate dense smoothing.
Fig. 6: Empirical coverage versus interval sharpness for learned trajectory ensembles. Each point is a seed–regime–noise block; the dashed line is nominal 90% coverage. Calibration is stable on test, while direct RTS-residual models under-cover under shift.
Split
Estimator
Modal RMSE (km)
Best-of-16 (km)
Energy
Mode recall
Traj. 90% cov.
Test
Batch MAP
13.39
13.39
0.00773
0.500
0.940
Test
Multistart
13.53
11.09
0.00735
0.664
0.933
Test
EKF–RTS
19.33
25.54
0.00907
0.619
0.918
Test
UKF smoother
19.52
26.72
0.00919
0.622
0.924
Test
Physics diffusion
19.33
12.60
0.00939
0.870
0.923
Blind
Batch MAP
13.51
13.51
0.00780
0.500
0.869
TABLE V: Nonlinear Multimodal Results (Means Over 100 Seeds, Three Regimes, and Two Noise Levels)
Fig. 7: Proper scoring and catalog-mode recovery under nonlinear tracking. Diffusion has the highest mode recall, while multistart retains the best mean energy on both test and blind splits.
Fig. 8: Custody false acceptance and selective-risk AUC. Lower is better on both axes. The diffusion ensemble does not dominate the classical references.
Width
Steps
Test
Blind
Energy
Best-16 (km)
False accept
Energy
Best-16 (km)
False accept
160
1800
0.010083
13.490
0.0556
0.007413
11.787
0.0343
160
7200
0.010103
13.295
0.0553
0.007506
11.668
0.0347
320
1800
0.010144
13.604
0.0556
0.007471
11.904
0.0348
320
7200
0.010122
13.372
0.0552
0.007557
11.773
0.0342
640
1800
0.010201
13.819
0.0545
0.007537
12.173
0.0352
TABLE VI: Fresh-Split Scaling Results (Means Over 100 Seeds)
Fig. 9: Fresh-split capacity–budget scaling. Solid circles denote test and dashed squares blind. Additional updates modestly reduce best-of-16 error; increased width does not improve energy or custody and generally worsens set recovery.
Method
N
Mean custody [95% CI]
Failure fraction [95% CI]
Mean handoffs [95% CI]
Seed replay
1000
0.4697 [0.4513, 0.4879]
0.862 [0.840, 0.883]
2.670 [2.544, 2.797]
Bootstrap
1000
0.4609 [0.4422, 0.4797]
0.862 [0.841, 0.883]
2.742 [2.609, 2.877]
Force-tier GMM
1000
0.5177 [0.5003, 0.5356]
0.848 [0.825, 0.870]
2.507 [2.401, 2.612]
Constrained diffusion
1000
0.4631 [0.4444, 0.4817]
0.861 [0.840, 0.882]
2.690 [2.562, 2.821]
TABLE VII: GoDSAT-Compatible Monte Carlo Results by Input Population
Graduate Student, Department of Aeronautics and Astronautics, Massachusetts Inst. of Technology, Cambridge MA · Associate Professor, Department of Aeronautics and Astronautics, Massachusetts Inst. of Technology, Cambridge MA