PathFM: Fast Generative Forecasting of Multivariate Trajectories

• Point and marginal scores do not capture trajectory-dependent properties such as cumulative demand, peak load, and threshold-crossing times.Source: §1 · pp. 1–2
• Marginal quantiles can approximate multimodal or sharply peaked distributions roughly, while leaving temporal dependence unspecified.Source: §1 · pp. 1–2
• Existing generative time-series models can represent joint paths, but the cited models have not combined high marginal accuracy with efficient sampling.Source: §1 · p. 2
• PathFM uses conditional flow matching for complete future paths, with distillation, cached history attention, and a calibration loss to improve sampling speed and marginal scores.Source: §1 · p. 2
• PathFM ranks second on fev-bench marginal scores and first on fps-bench path scores.Source: §1 · p. 2
• The introduction presents fps-bench as a trajectory-quality benchmark and reports PathFM’s high forecasting accuracy on chaotic systems.Source: §1 · p. 3

• A trajectory generator represents the predictive law as X = Gθ(I, Z), with each noise draw producing one complete future path.Source: §2 · p. 3
• Coordinatewise quantiles and marginal parameters leave temporal and cross-variable dependence unspecified, so they do not determine autocorrelation or distributions of sums, maxima, and path-dependent events.Source: §2 · p. 3
• Existing benchmarks report point and marginal scores, while fps-bench adds evaluation of temporal dependence and trajectory-dependent quantities.Source: §2 · p. 3

3 FAST AND CALIBRATED TRAJECTORY FORECASTING

• PathFM learns a flow over complete future trajectories, then distills it for faster sampling and refines its marginal scores.Source: §3 · p. 4

3.1 PRETRAINING BY FLOW MATCHING

• Conditional flow matching trains a velocity field to transport source noise toward the conditional law over complete futures, given history, masks, and known future covariates.Source: §3.1 · p. 4

3.2 EFFICIENT BACKBONE FOR JOINT GENERATION

• Late entry and history-only attention keep history representations independent of sampled paths, allowing history attention values to be reused across paths and flow evaluations.Source: §3.2 · p. 5

3.3 POST-TRAINING: DISTILLATION WITH CALIBRATION

• Parallel decoding distillation groups flow-time intervals so one or two backbone evaluations can replace a long rollout; calibration backpropagates fair CRPS while retaining PDD teacher supervision.Source: §3.3 · pp. 5–6

3.4 INFERENCE: ENSEMBLE SAMPLING

• At inference, PathFM stratifies initial Gaussian noise and propagates each draw through the distilled sampler to generate complete trajectories.Source: §3.4 · p. 6

• fps-bench combines established scoring rules in a common protocol for comparing finite trajectory ensembles, because marginal accuracy alone cannot establish temporal dependence or trajectory uncertainty.Source: §4 · p. 6
• The benchmark uses energy, differenced energy, variogram, and scalar CRPS scores for horizon totals, maxima, and first-passage times across 94 fev-bench tasks.Source: §4 · p. 6
• Quantile forecasts are completed under independent or comonotone rank assumptions, preserving their interpolated marginals while allowing trajectory scores to vary with the dependence assumption.Source: §4 · p. 6

5.1 EXPERIMENTAL SETUP

• The evaluation covers fev-bench and TIME marginal accuracy, and compares 32-path PathFM forecasts with quantile models’ better independent or comonotone completions.PathFM uses stratified Gaussian sources and one model evaluation per output block; quantile models are evaluated under both completions.Source: §5.1 · p. 7

5.2 RESULTS

• PathFM-300M and PathFM-126M achieve the two highest FPS scores while remaining competitive on marginal accuracy; PathFM improves on Sundial’s native forecasts across all three benchmarks.PathFM-300M ranks second on fev-bench and third on TIME, while both sizes outperform TimesFM-3 and Chronos-2 on fps-bench.Source: §5.2 · p. 7, Figure 3 · p. 7
• PathFM’s clearest gains over TimesFM-3 are in temporal-change, horizon-total, and maximum scores, highlighting trajectory quality beyond marginal forecasts.These FPS components address downstream tasks that depend on temporal structure.Source: §5.2 · p. 7

5.3 ACCURACY, MODEL SIZE, AND INFERENCE COST

• With one evaluation per output block, PathFM generates 32 paths in 51.6 ms on an NVIDIA L4 and lies on the observed accuracy–latency frontier for path forecasts.The model combines the highest FPS skill among the compared models with a 126M-parameter network.Source: §5.3 · p. 8, Figure 4 · p. 8
• On all 100 fev-bench tasks, 32-path stratified sampling achieves almost the same FEV skill as 256 iid paths; the two-evaluation student slightly exceeds its ten-evaluation teacher.With one evaluation, the student approaches the teacher’s FEV skill.Source: §5.3 · p. 8, Figure 5 · p. 8
• PathFM-300M reaches lower training and corpus-evaluation losses than PathFM-126M at the same update count and improves marginal accuracy and trajectory quality, at roughly twice the inference latency.The scaling curves suggest larger models may continue to improve accuracy.Source: §5.3 · p. 8, Figure 6 · p. 9

6 FORECASTING CHAOTIC DYNAMICS

• The evaluation compares fine-tuned Chronos-2, PANDA, and PathFM on point forecasts and long continuations of 135 DYSTS systems.Point accuracy is assessed at horizons 64, 128, 256, and 512; continuation diagnostics cover invariant-distribution distance, correlation dimension, power spectra, and segment similarity.Source: §6 · p. 9, §6.1 · p. 9

6.1 FORECAST ACCURACY AND DYNAMICAL FIDELITY

• PathFM achieves better dynamical fidelity over 4,096-step rollouts than all evaluated baselines, reduces PANDA’s four long-term diagnostic scores by 49–69%, and has the lowest pointwise error at every reported horizon.The gains persist when PathFM is scaled down to approximately PANDA’s 72M parameter count.Source: §6.1 · p. 9, Table 2 · p. 9
• PathFM improves across every reported metric relative to Chronos-1 and Chronos-2, combining stronger point accuracy with greater dynamical fidelity.Chronos-1 has lower W1 and PSD H2 than Chronos-2, but higher pointwise, correlation-dimension, and segment-analog errors.Source: §6.1 · p. 9
• Distillation increases point-forecast error relative to the ten-step sampler, but one- or two-step PathFM still outperforms PANDA on point accuracy and all four dynamical diagnostics.The paper hypothesizes that sensitivity to perturbations amplifies small distillation errors during autoregressive forecasts.Source: §6.1 · pp. 9–10

6.2 WHAT SAMPLING PRESERVES

• In the Lorenz example, PathFM’s sampled path retains a two-lobe portrait and sustained x variability while plotted baselines collapse within 256 steps; for Halvorsen, it retains a broad three-loop form.PANDA follows a smaller recurrent orbit in the Halvorsen example.Source: §6.2 · p. 10, Figure 7 · p. 10

7 DISCUSSION AND CONCLUSION

• The paper’s stated conclusion is that PathFM’s strongest gains concern complete-trajectory quality and the downstream predictions those trajectories support.In the dynamical study, independently continued paths preserve variability that point forecasts can lose.Source: §7 · p. 10
• PathFM does not yet support conditioning on modalities beyond time series, and its four chaotic-system diagnostics remain well above the held-out-truth reference.Accurately recovering the invariant distribution remains an open challenge.Source: §7 · p. 10

AI USE STATEMENT

• The authors report using generative AI for dataset work, implementation, analysis, literature retrieval, drafting, and editing, and say they reviewed the assisted work and take responsibility for the final content.Source: AI USE STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

• The authors say they will release code and model weights to support reproducibility.Source: REPRODUCIBILITY STATEMENT · p. 11

A RELATED WORK

• PathFM joins prior work on marginal forecasts and generative trajectory models, while using complete multivariate paths to support both marginal summaries and trajectory-dependent quantities.Source: A RELATED WORK · p. 14

A.1 CONCURRENT WORK

• The concurrent t0-gamma, QiYao-M, and EXAONE 1.0 models predict marginal quantiles, whereas PathFM directly generates joint multivariate trajectories.Source: §A.1 · p. 14

B MODEL AND TRAINING

• The model section covers the 126M and 300M configurations, training recipe, distillation, calibration, architecture controls, benchmarks, and inference cost.The flow model has 125.1M parameters; its distilled version has 126.3M including PDD heads.Source: B MODEL AND TRAINING · p. 15

B.1 BACKBONE AND INPUT

• The general-domain backbone uses velocity prediction, ReLU, rotary temporal positions, dynamic adaptive RMS normalization, and normalized temporal and channel attention.The 126M and 300M backbones have 12 and 30 blocks, respectively.Source: §B.1 · p. 15

B.2 TRAINING RECIPE

• Both model sizes use a four-stage schedule; PDD initializes from the terminal teacher’s EMA weights, and calibration resumes the PDD student and its optimizer and EMA state.Training uses real-world and synthetic series and samples contexts up to 8,192 steps and horizons up to 480 steps.Source: §B.2 · pp. 15–16, Table 3 · p. 15

B.3 DISTILLATION AND EVALUATION BUDGET

• At one evaluation, PDD has higher skill scores than the ten-evaluation teacher in both model families, and additional evaluations give no consistent improvement.Figure 8 compares teacher–student pairs at 1, 2, 4, and 10 evaluations, reporting FEV and six-component FPS skill.Source: §B.3 · p. 16, Figure 8 · p. 16

B.4 ARCHITECTURE AND TRAINING ABLATIONS

• Stratified sampling raises empirical FPS from 0.5248 to 0.5329, with a paired 95% task-bootstrap interval of [0.0042, 0.0131] for the skill gain.The cumulative study also reports that distillation improves both scores while reducing evaluations from ten to one.Source: §B.4 · p. 17, Table 4 · p. 17

B.5.1 MODEL SIZE AND FORECAST ACCURACY

• The 300M model improves on the 126M model in FEV and FPS skill and GIFT-Eval and TIME CRPS, ranking second on FEV, third on TIME, and first on FPS among baselines.Figure 9 compares nine models across FEV SQL skill, TIME normalized CRPS, and FPS path skill.Source: Figure 9 · p. 17, §B.5.1 · p. 18
• Against external baselines, the 126M model ranks fourth and the 300M model third on GIFT-Eval CRPS under the stated full-configuration evaluation.The evaluation uses all 97 configurations without series or window caps and with cross-learning disabled.Source: §B.5.1 · p. 18, Table 5 · p. 18

B.5.2 ACCURACY AND INFERENCE COST

• Both PathFM variants lie on the observed accuracy–latency frontier for path forecasts in the 300M timing comparison.Figure 10 reports median time to return nine quantiles or forecast paths for one query on an NVIDIA L4 at context 2,048 and horizon 288.Source: §B.5.2 · p. 18, Figure 10 · p. 18

C.1 FINITE-ENSEMBLE APPROXIMATION

• Finite ensembles introduce Monte Carlo variation in forecast summaries and scores, separate from model error.The section distinguishes finite-ensemble approximation error from model error before describing ensemble construction.Source: §C.1 · p. 19

C.1.1 FORECAST QUANTITIES AND MODEL ERROR

• For an iid sample average of a fixed finite-variance path functional, Monte Carlo standard error scales as m^-1/2; rare-event estimates can remain demanding at small ensemble sizes.Increasing sample count reduces sampling error but does not remove model error; dependent or stratified draws require a different variance analysis.Source: §C.1.1 · p. 19

C.1.2 ENSEMBLE CONSTRUCTION AND STRATIFICATION

• The ensemble construction stratifies Gaussian source values by assigning each rank once per coordinate, with independent permutations that do not tie ranks across times or channels.Source: §C.1.2 · p. 19

C.2 MARGINAL COVERAGE AND ENSEMBLE SIZE

• Increasing iid ensemble size from 32 to 1,024 raises 80% interval coverage by 4 percentage points across all 100 fev-bench tasks, with diminishing gains.Figure 11 plots marginal 80% interval coverage against sample count and references Chronos-2 and TimesFM-3 native percentile outputs.Source: §C.2 · p. 20, Figure 11 · p. 20

C.3 INFERENCE LATENCY

• PathFM returns an ensemble in 51.6 ms for one history and 215.2 ms for sixteen; at both batch sizes its one-evaluation version is faster than Sundial and TimesFM-3.Chronos-2 and TiRex-2 have lower latency, while PathFM achieves higher FPS skill.Source: §C.3 · p. 20, Table 6 · p. 20

C.3.1 INFERENCE SPEEDUP BREAKDOWN

• Packing and distillation reduce the packed teacher’s 2,663.8 ms latency to 201.9 ms for the two-evaluation student, while packing cuts peak allocated memory from 3.22 to 1.58 GiB.Deeper history K/V caching changes teacher latency only to 801.4 ms and is disabled for the student.Source: §C.3.1 · p. 20, Figure 12 · p. 21

D EVALUATION METRICS

• The empirical CRPS and energy score evaluate the finite forecast distribution without assuming ensemble members are independent; the variogram component evaluates moments of that same empirical forecast.The evaluation section defines temporal scores and aggregation rules alongside these empirical forecast scores.Source: §D.1.1 · p. 21, §D.1.2 · p. 21

D.1.2 TEMPORAL PATH SCORES

• The first-passage score requires a fully observed future and finite threshold, records forecast steps, and codes no crossing as L + 1; it therefore evaluates a censored event time.Sum, maximum, and first-passage scores identify their respective functional distributions rather than the full path law; the cross-variable variogram compares channel pairs at the same forecast time.Source: §D.1.2 · p. 22, §D.1.3 · p. 22

E.1 TEMPORAL FORECAST ACCURACY

• PathFM achieves the highest skill on all six temporal FPS components, with a 10.6% average improvement over TimesFM-3, the strongest baseline.Quantile baselines’ independent completion usually gives better temporal scores than comonotone completion, though comonotone completion can improve sum skill.Source: §E.1 · p. 22, Figure 13 · p. 23, Table 8 · p. 23

E.3 FITTED GAUSSIAN TEMPORAL COPULAS

• The fitted copula estimates temporal dependence from historical forecast errors and uses a stationary latent AR(1) process to generate dependent ranks for baseline quantiles.It estimates a shared lag-one coefficient from four pre-test windows, clips it to [−0.95, 0.95], and uses task-pooled estimates when item information is insufficient.Source: §E.3 · p. 24
• Copula fitting raises FPS for Chronos-2, TimesFM-3, and Toto-2.0-2.5B versus independent completion, while PathFM has a higher FPS score without task-specific fitting.Table 10 covers 94 common fev-bench tasks.Source: §E.3 · p. 25, Table 10 · p. 25

F CHAOTIC SYSTEMS

• The chaotic-system experiments examine how model size, sampling, and fine-tuning affect point accuracy and long-run dynamics.Training uses the PANDA-72M reproduction corpus, with ten parent systems reserved and the remaining trajectories split into training and validation sets.Source: F CHAOTIC SYSTEMS · p. 25, §F.1 · p. 25

F.1 SETUP AND FORECASTING PROTOCOLS

• Evaluation includes point forecasts and long continuations, with distinct protocols for model-progress checks, forecast horizons, and dynamical diagnostics.Point forecasts use six 512-context windows from each of 9,347 trajectories; long-continuation diagnostics use 4,096-step paths across 135 systems.Source: §F.1 · pp. 25–26

F.2 MODEL SIZE AND SAMPLER CHOICE

• The 75M PathFM has lower errors than 72M-parameter PANDA at all four forecasting horizons and on all four dynamical diagnostics.The comparison uses ten Euler evaluations per block and reports lower values as better.Source: §F.2 · p. 26, Table 11 · p. 26
• For 126M long continuations, two-evaluation PDD reduces mean W1 from 0.522 with two-step Euler to 0.316, approaching ten-step Euler’s 0.251.At one or two evaluations, PDD also improves point accuracy over Euler, although ten-step Euler remains best at every horizon within each size.Source: §F.2 · p. 26, Figure 14 · p. 26, Table 12 · p. 27

F.3 FULL CONTINUATIONS

• Across the displayed 4,096-step examples, PathFM sustains Lorenz’s two-lobe motion and Rössler’s varying oscillation amplitudes, while Chronos-2 and PANDA show collapse or altered dynamics.The figure uses the same initial conditions across panels; its shading marks the first 512 forecast steps.Source: §F.3 · p. 28, Figure 15 · p. 28

F.4 METRIC DEFINITIONS

• The diagnostic measures distance to an empirical state distribution rather than an exact invariant measure; its reference cloud retains at most 20,000 states, with each cloud subsampled to at most 10,000.The pooled target uses all ten original trajectories, including the test trajectory, and its own standardization; it is a descriptive geometric reference, not an independent held-out estimate.Source: §F.4 · p. 29