FlowATC: Aircraft Trajectory Prediction via Flow Matching

I. Introduction

• A single observed history can correspond to several physically plausible futures, making a distribution over futures more relevant than a single point estimate for conflict assessment.Position data alone may not reveal whether an aircraft will turn, continue, or follow another maneuver.Source: §I · p. 2
• FlowATC uses a DiT-style Transformer backbone with DDPM or CFM to frame trajectory prediction as token-level inpainting.The architecture exploits the Transformer backbone’s compatibility with both generative objectives.Source: §I · p. 3

Related work

• Prior aircraft-prediction work differs in setting and representation: FlowATC targets low-altitude terminal airspace, native irregular ADS-B timing, and absolute coordinates without chart supervision.The comparison is with Figuet et al.’s en-route Swiss Free Route Airspace study, which uses normalized aircraft-centric coordinates and resampled data.Source: Related work · p. 4
• At matched capacity, CFM is more accurate than DDPM, and FlowATC reports recovering flown maneuvers and near-calibrated next-maneuver distributions at branch points.The contribution list also identifies constant-velocity, deterministic LSTM, and CVAE baselines in the benchmark.Source: Related work · p. 4

• ADS-B broadcasts aircraft state, including position, altitude, velocity, and vertical rate, and can provide high-coverage, low-cost trajectory data over busy terminal airspace.The states are derived primarily from onboard GNSS and received by ground stations.Source: A. Aircraft Trajectory Data · p. 4
• The ADS-B collection ran for 12 days and yielded 21,515,794 raw state vectors within a 60-nautical-mile-radius geofence centered on San Francisco Bay.The geofence and referenced airports are shown in Fig. 1.Source: A. Aircraft Trajectory Data · pp. 4–5, Figure 1 · p. 5
• The collection includes large commercial aircraft at 41.8%, light general aviation at 32.1%, heavy aircraft at 10.8%, and other lighter traffic at 15.3%.Source: A. Aircraft Trajectory Data · p. 5

B. Trajectory processing

• Each 86-point sequence is split into a 43-point observed prefix and a 43-point future suffix, giving an average forecast horizon of roughly 128 seconds.The two-minute lookahead was identified by pilots as operationally relevant.Source: B. Trajectory processing · p. 5
• The six-dimensional trajectory representation uses local SFO-centered Cartesian position and velocity, allowing geographic position to encode location-specific structure.Features are independently z-score normalized using training-split statistics.Source: B. Trajectory processing · p. 5

III. Evaluation

• Because multiple maneuvers may be acceptable for one observed prefix, the evaluation includes best-of-K ADE/FDE and KDE negative log-likelihood.The evaluation section describes distributional assessment as difficult in practice and builds on Trajectron++ [4].Source: §III · p. 6

A. Average and Final Displacement Error

• ADE measures how close a sampled trajectory is to ground truth, while FDE measures the position error at the final future step.The horizontal position is represented as p_n = (x_n, y_n); vertical position is treated separately because its variance is secondary to horizontal variations.Source: A. Average and Final Displacement Error · pp. 6–7
• The final prediction token is on average 128 seconds ahead, with the central 80% of windows spanning 95 to 165 seconds because ADS-B sampling is irregular.Source: A. Average and Final Displacement Error · p. 7

B. Best of K

• Best-of-K minADE@K and minFDE@K address multimodal futures by rewarding whether at least one sample lies close to ground truth.The paper reports K values of 1, 5, and 20 to quantify how quickly added samples improve coverage.Source: B. Best of K · p. 7

C. Density Calibration

• Best-of-K metrics can give the same minADE@K to a cloud with one on-target sample and arbitrary remaining samples as to a cloud tightly bracketing the truth, while KDE-NLL evaluates the full distribution.The latter can yield a density usable for conflict detection.Source: C. Density Calibration · p. 7
• At each future step, the method fits a Gaussian KDE with Scott’s-rule bandwidth to 50 independent completions’ horizontal positions.The empirical covariance and Scott factor define the bandwidth.Source: C. Density Calibration · pp. 7–8
• In Fig. 2, hyper-diffuse samples have minFDE@20 comparable to the calibrated multimodal case but markedly lower likelihood, while a sharp displaced cloud makes NLL diverge.The calibrated bimodal case scores well against either plausible outcome: NLL_A=15.5 and NLL_B=14.9 nats.Source: Figure 2 · p. 8
• NLL is averaged over the test set, lower is better, and is evaluated at future steps 10, 20, and 43; density is estimated horizontally, with the vertical axis handled separately.Source: C. Density Calibration · p. 9
• The finite-K KDE log score is approximate rather than exactly proper, and a 50 m per-dimension bandwidth floor prevents a collapsed cloud from driving NLL to −∞.The score reflects sharpness and calibration, rewarding concentrated mass but penalizing overconfidence that excludes the ground truth.Source: C. Density Calibration · pp. 8–9

IV. Baseline Architectures

• The baselines span kinematic, deterministic-neural, and probabilistic-generative families, all using the same 43-point observed prefix and 43-point future suffix for like-for-like comparison.Source: §IV · p. 9

A. Constant Velocity

• Constant velocity propagates the last observed state using target timestamps to account for irregular ADS-B sampling; it requires no training and is the natural lower bound for structure-aware models.Its deterministic best-of-K metrics are constant in K.Source: A. Constant Velocity · p. 9

B. Deterministic LSTM Encoder–GRU Decoder

• The deterministic baseline encodes 43 observed tokens with an LSTM and autoregressively decodes 43 future states with a GRU, trained on per-step displacement MSE and integrated into positions.Elapsed time is included so the rollout can account for irregular ADS-B sampling; its single output makes @K metrics collapse to the deterministic result.Source: B. Deterministic LSTM Encoder–GRU… · p. 9

C. Conditional Variational Autoencoder (Trajectron++)

• The CVAE encodes the observed past, samples a latent future mode, and decodes trajectories; independent latent samples provide diverse trajectories for best-of-K evaluation and density estimation.It serves as the probabilistic non-diffusion baseline and follows Trajectron++ [4].Source: C. Conditional Variational Autoencoder… · p. 10

A. Problem Formulation

• Because unobserved runway assignments, controller instructions, and sequencing make future trajectories multimodal, the target is a conditional sampler rather than a point estimate.Under mean-squared error, deterministic predictors converge to the conditional mean, which can average distinct modes into a physically implausible trajectory.Source: A. Problem Formulation · p. 10
• For fixed Xobs, Conditional Flow Matching regresses a velocity field whose unique population minimizer, integrated exactly from x(0) ~ N(0,I), has terminal marginal exactly p(·|Xobs).Source: A. Problem Formulation · pp. 10–11

B. The DiT Backbone

• The DiT backbone injects flow time through Adaptive Layer Normalization and uses block-causal attention so future tokens attend to the observed prefix, but observed tokens remain independent of future noise.Future tokens attend freely to one another, supporting full bidirectional attention within the future block.Source: B. The DiT Backbone · p. 11

C. Trajectory as an Image: Inpainting the Future

• Each trajectory is represented as an 86-token sequence with 43 observed tokens and 43 noisy future tokens inpainted from Gaussian noise.The observed past enters through concatenated context tokens without a separate encoder; extra signals can also be appended as context tokens.Source: C. Trajectory as an Image: Inpainting… · pp. 11–12

VI. Results

• Sampling K completions yields trajectory fans and final-point occupancy estimates for downstream conflict-risk estimation, with wider densities around uncertain approach-turn initiation.The reported high-density region brackets the ground-truth final position; straight en-route segments have narrow, elongated densities.Source: §VI · p. 13, Figure 6 · p. 14

A. Model Comparison

• At K=20, Flow Tiny and Diffusion Tiny achieve 41% and 31% lower minADE@20 than CVAE, respectively.The comparison is at matched capacity of 1.5 M parameters on the held-out test set.Source: A. Model Comparison · p. 15, Table 3 · p. 15
• At 43 steps, CVAE NLL@43 is 31.7 versus 18.1 for Flow Tiny, corresponding to roughly 8 × 10^5 times lower density at the realized endpoint.CVAE best-of-20 samples can land near the ground truth even though its full predicted density is comparatively poorly calibrated.Source: A. Model Comparison · p. 15
• Because capacity is held fixed, CFM’s consistent advantage over DDPM points to the training objective, but the comparison uses single runs and does not quantify seed variance.The authors revisit this limitation at scale in Section VI.B.Source: A. Model Comparison · pp. 15–16
• At K=1, the deterministic LSTM has lower vertical error than Flow Tiny (78.4±0.4 m versus 94.2±0.5 m).A single stochastic draw inherits the predicted distribution’s spread rather than estimating its mean.Source: Table 4 · p. 16, A. Model Comparison · p. 16
• At K=20, Flow Tiny’s vertical error is 25.0±0.2 m, versus 28.3±0.2 m for Diffusion Tiny and 43.8±0.3 m for CVAE.This is a 43% reduction over CVAE, close to the 41% measured horizontally.Source: Table 4 · p. 16, A. Model Comparison · p. 16
• In a direct comparison, CFM (Flow Tiny) beats DDPM (Diffusion Tiny), with 15% lower minADE@20 (390.5 vs. 459.9), 17% lower minFDE@20, and consistently lower NLL at all three horizons.Source: A. Model Comparison · p. 15
• At 10 steps, the three generative models have similar NLLs (13.4–14.6 nats), but their NLLs diverge sharply by 43 steps.CVAE’s calibration curve rises steeply after n≈20, while Diffusion Tiny and Flow Tiny remain nearly flat.Source: A. Model Comparison · p. 15, Figure 7 · p. 16
• At a two-minute horizon, altitude is dominated by the discrete choice to level off, climb, or descend, a branching that a single point estimate cannot represent.The vertical-error results reproduce the horizontal narrative rather than adding a new one.Source: A. Model Comparison · p. 16

B. Scaling

• At matched capacity, CFM beats DDPM on minADE@20 by 15% (Tiny), 26% (Small), and 11% (Large), with the advantage narrowing at Large.The comparison measures minADE@20 reduction relative to DDPM at matched capacity.Source: Table 5 · p. 17, B. Scaling · p. 17
• Flow Small reaches 306.1 m minADE@20 versus 307.4 m for Flow Large at 67.0 versus 157.1 ms per request, a 2.3× latency advantage.The authors suggest CFM approaches a data-limited regime around 7 M parameters on this 12-day dataset.Source: B. Scaling · p. 17, Table 5 · p. 17, Table 6 · p. 20
• KDE NLL@43 is best for Flow Tiny at 18.1 nats and is 20.2 nats for Flow Small and Large, even as displacement error improves with scale.The authors distinguish best-of-K accuracy from distributional calibration; scaling improves the former without guaranteeing the latter.Source: B. Scaling · p. 17

C. Extrapolation to Future Timesteps

• Extrapolation queries prediction dates beyond training offsets with a fixed 43-token future block; beyond roughly 150 s, comparisons rely on interpolated continuations, with only 16% of windows natively reaching 150 s.Ground truth beyond the observation window is obtained by cubic-spline interpolation of continuing ADS-B trajectories.Source: C. Extrapolation to Future Timesteps · pp. 17–18
• At K=1, diffusion models degrade faster than CVAE at long horizons, but with K≥5 Flow is most accurate across the horizon; only CVAE K=1 overtakes it beyond about 200 s.The reported comparison attributes that CVAE displacement-error result to a single sampled trajectory, not better uncertainty calibration.Source: C. Extrapolation to Future Timesteps · p. 18
• From about 150 s onward, Flow and Diffusion KDE-NLL plateaus near 20 nats while CVAE diverges to about 60 nats by 360 s.The paper interprets CVAE’s lower displacement error beyond about 200 s as a single-sample effect rather than a calibrated uncertainty estimate.Source: C. Extrapolation to Future Timesteps · pp. 18–19

D. Operational Robustness

• Stride-2 retraining costs 4.9% in minADE@20 while roughly halving inference time; stride 8 costs 44.4% but runs 5.0× faster, with near-term NLL@10 essentially unchanged.Long-horizon calibration degrades more than displacement results alone suggest.Source: D. Operational Robustness · p. 19
• With shortened history, error stays essentially flat to about 21 retained points, rises sharply below about 15, and reaches 2.6× full-history error at two points.Because the frozen prefix is mildly out of distribution, the authors treat this as a conservative upper bound on the cost of genuinely shorter history.Source: D. Operational Robustness · p. 19, Figure 10 · p. 20

VII. Spatial Bias

• Without chart or procedure supervision, FlowATC reproduces Bay Area airspace structure, including approach turns, holding patterns, and descent profiles.The paper attributes this to latent structure encoded in absolute Cartesian coordinates.Source: §VII · p. 20

A. Identifying Recurring Turn Patterns From ADS-B Trajectories

• The catalogue’s association of turns with nearby navigation references indicates spatial proximity, not that a fix, instruction, or procedure caused a maneuver.The HLHRS_l1 name likewise denotes proximity and does not establish that HLHRS caused the maneuver or belongs to the corresponding published procedure.Source: A. Identifying Recurring Turn Patterns… · pp. 21–22
• The ADS-B-derived catalogue contains 433 recurring patterns and 32,765 turn events, including an SFO departure corridor broadly consistent with the NIITE FOUR procedure.The catalogue represents recurring local maneuver modes, but geometry and fix proximity alone do not identify the procedure assigned to an individual flight.Source: A. Identifying Recurring Turn Patterns… · pp. 21–22

B. Qualitative Evidence of Learned Spatial Bias

• When histories remained ambiguous, FlowATC sampled multiple geographically recurring continuations, covered ground-truth futures, and favored historically dominant branches.The sampled modes included northbound and eastbound departures near SFO and several branches plus straight continuation near Daly City.Source: B. Qualitative Evidence of Learned… · p. 23

C. Probabilistic Reproduction of maneuvers at Branch Points

• Across 28 branch points, FlowATC placed probability mass on observed maneuver modes at comparable angles and weights, matching straight-flight shares within a few percentage points.The evaluation covered 2,329 held-out passages, including aircraft that did not turn; for catalogued turns, the heading-change measure recovered the catalogued angle with a median error of 1.0°.Source: C. Probabilistic Reproduction of… · p. 24
• 3.3° pooled 1-Wasserstein distance versus about 3.8° for an ideal history-blind sampler; FlowATC reaches that level at 16 of 28 branch points.It predicts a straight-through share of 32.2% against 32.9% observed, with 0.99 correlation across branch points.Source: C. Probabilistic Reproduction of… · p. 25, Figure 14 · p. 25
• 47% CRPS reduction versus the history-blind sampler (95% bootstrap interval 36–62%), with better scores on 82% of passages and at all 28 branch points.The central 80% and 95% sample intervals contain observed outcomes for 81% and 92% of passages, respectively, indicating slightly narrow tails.Source: C. Probabilistic Reproduction of… · pp. 25–26, Table 7 · p. 26
• Flow Small achieves 41% CRPS skill versus Flow Large’s 47%, while Flow Tiny and the CVAE reach only 25% and roughly triple the marginal distance.The CVAE is the only model with clearly too-narrow intervals: its central 95% interval covers 84% of outcomes.Source: C. Probabilistic Reproduction of… · p. 26
• Drawing from cataloged turn angles alone performs worse than the history-blind sampler because it describes only aircraft that turn.Source: C. Probabilistic Reproduction of… · p. 26

VIII. Conclusion

• Best-of-K accuracy and distributional calibration are distinct; K independent samples turn single-sample weaknesses into calibrated spatial densities for downstream conflict-risk estimation.Source: §VIII · p. 26
• At matched capacity, CFM and DDPM surpass CVAE by 31–41% on minADE@20, and CFM beats DDPM by 11–26% on minADE@20.Source: §VIII · p. 26
• Stride 2 costs under 5% in minADE@20 while roughly halving inference time; stride 8 gives a 5× speedup at a 44% accuracy cost.Source: §VIII · p. 27
• Beyond the training horizon, FlowATC remains the most accurate model up to 360 s with several samples, and its calibration stays stable where CVAE calibration diverges.Source: §VIII · p. 27
• Without chart or procedure supervision, FlowATC reproduces maneuver distributions at 28 branch points, including the third of aircraft that continue straight, and improves on a history-blind sampler by 47% in CRPS.Source: §VIII · p. 27

Future Work

• Future work proposes adding ATC voice, weather, and FAA airspace structure as context, with each modality expected to reduce density variance at decision points.Source: Future Work · p. 27

B. Data Preprocessing details

• Preprocessing projects geodetic positions into an SFO-centered tangent plane to address nonlinear longitude, circular track, and incompatible scales; elapsed sample times receive sinusoidal embeddings.Source: B. Data Preprocessing details · p. 30

A. Random Sample of Flow-Large Predictions on the Test Set

• In an uncurated 40-window sample, most Flow Large densities bracket ground truth, including a reproduced holding pattern, while 2 of 40 miss it.Source: A. Random Sample of Flow-Large… · p. 30, Figure 15 · p. 31

C. Flow matching and DDPM

• CFM learns a velocity field along straight-line paths from Gaussian noise to data, with a simpler loss and fewer inference steps than DDPM.At interpolation x_t = (1 − t)x_0 + t x_1, the target velocity is x_1 − x_0; the learned field is integrated at inference to produce a sample.Source: C. Flow matching and DDPM · p. 32

D. CVAE Decoder Variance at the First Prediction Step

• At step 1, CVAE’s 605 m decoder noise dwarfs its 111 m mean error; disabling noise changes minADE@1 from 542.6 m to 118.1 m and minADE@20 from 508.2 m to 853.4 m.The conditional latent is nearly one-hot, so removing decoder noise also removes much of the diversity across K draws.Source: D. CVAE Decoder Variance at the First… · pp. 32–33

E. Full K∈{1, 5, 20} Displacement Sweep

• Figure 17 provides the complete K∈{1, 5, 20} displacement sweep, adding K=5 to the main-text figure without a qualitatively new story.The main-text figure omits K=5 to keep three panels; the appendix includes the intermediate panel for completeness.Source: Figure 17 · p. 33, E. Full 𝐾∈{1, 5, 20} Displacement Sweep · p. 33

F. ADS-B Downsampling — Full Results

• NLL@43 rises from 20.2 to 22.2–23.0 nats across strides, while NLL@10 is essentially unchanged and reaches 12.9 vs. 13.3 nats at stride-8.The matched-horizon comparison shows that the long-horizon density loss is more substantial than the nat change alone suggests; the architecture nevertheless retrains effectively on lower-rate feeds.Source: F. ADS-B Downsampling — Full Results · pp. 33–34, Table 8 · p. 34

G. Full Extrapolation Sweep and Shard-Stitching Detail

• Extrapolation combines three independently resampled shards, assigning each time band to the run with the finest temporal resolution; visible seams reflect estimator changes, not model behavior.The bands are (0, 90] s, (90, 180] s, and (180, 360] s, using H_max values of 90, 180, and 360 s, respectively.Source: G. Full Extrapolation Sweep and… · p. 34

H. Turn Catalogue: Detection, Classification, and Grouping Protocol

• Turn events are transitions between stable ground-track segments; retained turns are grouped by local area and similar incoming, outgoing, and signed heading changes.Smaller course changes are retained only when path geometry supports them, and only recurring, geometrically coherent groups remain in the catalogue.Source: Figure 19 · p. 35, H. Turn Catalogue: Detection,… · p. 35

Passages

• Candidate passages must be airborne, aligned with and approaching a branch point, and predicted to reach it in 15–90 s; nearby windows from one aircraft are grouped into a passage.The evaluation retains 28 branch points with at least 40 remaining test passages.Source: §I · p. 36

Outcome

• On 771 test passages flying a catalogued turn, the predicted heading change matches its catalogued angle to a median of 1.0° and within 15° for 86%.The outcome is accumulated heading change at the first point after closest approach where the path is more than 4 NM from the branch point; an end-of-window value is used if it remains within the disc or never enters it.Source: §I · p. 36

Scores

• The score compares pooled sampled outcomes with observed outcomes, while the history-blind baseline samples other test passages at the same branch point.Bootstrap intervals resample branch points 1,000 times.Source: §I · p. 36