Time series foundation models are becoming increasingly popular for general-purpose forecasting. However, most leading models predict marginal quantiles, leaving the dependence structure of future trajectories unspecified. This distinction matters for trajectory-dependent quantities such as cumulative demand, peak load, and event timing, yet point and marginal scores do not reveal it. To model this dependence, we introduce PathFM, a trajectory-based forecasting model that uses flow matching to generate complete future paths. We further distill the model to one or two steps and share history attention cache across draws. Together, these enable PathFM to generate multiple paths for a single series in roughly the time needed for one query to a quantile model. The distilled sampler also makes it practical to optimize a calibration loss directly. PathFM achieves competitive marginal accuracy on the established forecasting benchmarks fev-bench and TIME. To complement these evaluations, we introduce Forecast Path Score Bench (fps-bench), which augments fev-bench tasks with energy and variogram scores and downstream metrics. PathFM achieves a higher aggregate path score on fps-bench than all evaluated external baselines. Experiments on chaotic systems further show that our flow model sustains long-run variability, while strong baselines rapidly collapse. PathFM makes generative time series forecasting both competitive and practical, enabling diverse downstream applications.
Figures & tables
Figure 1: Two advantages of trajectory generation. Left: a sampled Lorenz rollout sustains variability and a two-lobe structure, while the fine-tuned Chronos-2 trace contracts. Both use the dynamical models in Section 6 . Gray Lorenz curves show the observed continuation. Right: solar-output paths from PathFM and independent (ind.) and comonotone (com.) completions of the Chronos-2 (C2) quantiles. Horizontal dashed lines marks each solar path’s daily maximum; black denotes the observed future. In (g), boxes show medians and interquartile ranges across 32 paths, whiskers show the 10–90% range, and the dashed line marks the observed maximum.
Model
Forecast object
NFE
Task compatibility
Pretrained
Marginal
Path
Cross-variable
Quantile models
Chronos-2 ( Ansari et al., 2025 )
Marginal quantiles
1
✓
✓
×
×
TimesFM-3 ( Jain & Sen, 2026 )
Marginal quantiles
1
✓
✓
×
×
Toto 2.0 ( Khwaja et al., 2026 )
Marginal quantiles
1
✓
✓
×
×
Flow models
Table 1: Forecast objects, inference cost, and task compatibility of representative models.
Figure 2: Training, post-training, and sampling with PathFM. Flow matching learns a joint velocity field; distillation matches frozen-teacher velocities at student-proposed states. Calibration maps eight iid draws through one fused step of trainable Gθ to samples scored by fair CRPS, together with PDD supervision. Sampling uses stratified initial noise to improve coverage of the noise distribution with a small ensemble. Below: Flow enters after the flow-independent prefix, and history attention values are computed once and reused by separate packed future states.
Figure 3: Marginal accuracy and trajectory quality for PathFM-126M and PathFM-300M. Native samplers PathFM and Sundial are teal and black. FPS selects each quantile model’s better independent or comonotone completion.
Figure 4: Accuracy, latency, and model size across six forecasting models. Area denotes parameter count. Latency is the median time to return nine quantiles (a) or forecast paths (b) per query on an NVIDIA L4, with history and forecast lengths fixed across models. Quantile models use their better aggregate completion from Section 4 .
Figure 5: Ensemble size and distillation on fev-bench tasks. (a) Calibrated 126M model with different sampling strategies at NFE=1 ( Section C.1.2 ). (b) Its matched teacher and PDD student. Thin bars in (b) show pointwise 95% task-bootstrap intervals ( Section B.3 ).
Figure 6: Pretraining progress on general and chaotic time series. (a) Training velocity MSE; (b) EMA velocity MSE on fixed corpus windows for PathFM-126M and PathFM-300M; (c) EMA mean validation sMAPE for the 75M and 126M chaotic models at horizon 256, zoomed to 1M–2M updates.
sMAPE
Dynamics
Model
H=64
H=128
H=256
H=512
W1
PSD H2
CD err.
Seg. analog
Chronos-1 FT
24.80
31.86
39.69
47.24
0.411
0.148
0.750
16.40
Chronos-2 FT
21.07
26.78
33.28
40.07
9.686
0.269
0.569
11.12
PANDA
19.09
24.42
31.94
41.22
0.588
0.195
0.537
10.03
PathFM (Ours)
16.45
21.52
27.43
34.50
0.251
0.061
0.273
4.80
Held-out truth
—
—
—
—
0.047
0.014
0.055
1.00
Table 2: Point accuracy and dynamical fidelity (lower is better; best in bold).
Figure 7: Lorenz and Halvorsen continuations; PathFM uses one sampled path. Each column shows a full 4,096-step portrait above its first 512 x -coordinate steps. Gray curves show the observed continuation; open and filled circles mark the first and last forecast states.
Appendix figures & tables18 assets
Supplementary material from the paper’s appendix.
Appendix
Stage
Updates
Peak learning rate
Warmup
EMA decay
Flow matching
2,000,000
5×10−5
2,000
0.9999
Terminal decay (WSD)
100,000
5×10−5
—
0.9999
PDD
100,000
3×10−5
500
0.999
PDD + fair CRPS
100,000
10−5
—
0.999
Appendix
Table 3: General-corpus training schedule for both model sizes. Warmup is linear and included in the update budget. Flow matching and PDD use constant rates after warmup; WSD decays linearly to zero. The final stage decays linearly to zero for 126M and uses a constant rate for 300M. EMA starts at the end of warmup and is preserved across continuation stages. Each complete recipe contains 2.3M optimizer updates.
Figure 8: Teacher–student accuracy at different evaluation budgets. Rows show teacher–student pairs from the 126M and 295M families; columns show FEV and six-component FPS skill, with higher values preferred. NFE uses a logarithmic horizontal axis. Thin vertical bars show pointwise 95% task-bootstrap intervals from 2,000 resamples.
Cumulative change
NFE
FEV ↑
FPS ↑
R0
Flow matching with clean target
10
0.4314
0.4379
R1
+ Velocity prediction
10
0.4311
0.4216
R2
+ Time modulation, zero-init head, QK norm, weight decay
10
0.4468
0.4960
R3
+ History-only attention
10
0.4356
0.5008
R4
+ Two flow-free entry blocks
10
0.4381
0.5088
R5
+ Four packed training draws
10
0.4402
0.5076
Appendix
Table 4: Cumulative architecture and post-training study. Rows R0–R7 use iid sources.
Figure 9: Scaling PathFM from the 126M to the 300M family. Panels rank nine models on FEV SQL skill (100 tasks), TIME normalized CRPS (98 tasks), and FPS path skill (94 tasks). Native samplers PathFM and Sundial are teal and black. FPS selects each quantile model’s better independent or comonotone completion.
FEV
FPS
GIFT-Eval
TIME
Model
Skill ↑
Skill ↑
MASE ↓
CRPS ↓
MASE ↓
CRPS ↓
TimesFM-3
0.4900
0.4871
—
—
0.6398
0.5363
Chronos-2
0.4728
0.4699
0.6978
0.4854
0.6620
0.5563
TiRex-2
0.4550
0.4586
0.6973
0.4781
—
—
Toto-2.0-2.5B
0.4442
0.4261
0.6956
0.4759
0.6419
0.5394
TiRex
0.4258
0.4220
0.7158
0.4885
0.6831
0.5731
Appendix
Table 5: Forecasting accuracy across four benchmarks. Darker blue indicates better performance on a linear scale within each column; bold marks the best value. Dashes indicate unavailable results.
Figure 10: Accuracy, latency, and model size with the 300M PathFM variant. Circle area denotes parameter count. Latency is the median time to return nine quantiles (a) or forecast paths (b) for one query on an NVIDIA L4 at context 2,048 and horizon 288. Accuracy uses the native benchmark settings.
Figure 11: Marginal 80% interval coverage versus the number of forecast samples on fev-bench. Both model sizes use iid sampling. Horizontal references use the native 10th and 90th percentile outputs of Chronos-2 and TimesFM-3.
Nine quantiles (ms)
Paths (ms)
Model
Params. (M)
Batch 1
Batch 16
Batch 1
Batch 16
PathFM, one evaluation
126.3
51.7
215.3
51.6
215.2
PathFM, two evaluations
126.3
96.4
403.2
96.4
403.3
Chronos-2
119.5
21.8
66.6
22.5
72.9
TimesFM-3
330.7
160.0
508.2
160.6
515.1
Toto-2.0-2.5B
2,454.3
198.3
1,482.6
199.0
1,489.5
Appendix
Table 6: End-to-end latency on an NVIDIA L4 at context 2,048 and horizon 288.
Figure 12: Inference speedup breakdown on an A100 at batch 16, context 2,048, and horizon 480. The teacher uses ten Euler evaluations; the student uses two. All configurations use float32 weights with bfloat16 autocast.
Score
Distance or transformation
Energy (ES)
ρ(x,y)=∥x−y∥2
Differenced ES (DES)
ρ(x,y)=∥D(x)−D(y)∥2 , where D(x)=(x1,x2−x1,…,xL−xL−1)
Variogram (VS)
Squared errors in expected temporal contrasts, Equation 14
Sum CRPS
ρ(x,y)=∣∑txt−∑tyt∣
Max CRPS
ρ(x,y)=∣maxtxt−maxtyt∣
First-passage CRPS
ρ(u,v)=∣Tτ(u)−Tτ(v)∣ in original units, Equation 15
Appendix
Table 7: The definitions of the six temporal FPS components.
Figure 13: Component skills for PathFM and the three strongest baselines on fps-bench. Bars show skill relative to Seasonal Naive; higher is better.
Forecast
Completion
ES
DES
VS
Sum
Max
FPT
FPS skill
TimesFM-3
Independent
0.4913
0.3437
0.5998
0.5032
0.3966
0.5455
0.4871
TimesFM-3
Comonotone
0.4757
0.3650
0.4894
0.5049
0.3435
0.5462
0.4590
Chronos-2
Independent
0.4783
0.3255
0.5901
0.4797
0.3805
0.5224
0.4699
Chronos-2
Comonotone
0.4616
0.3590
0.4818
0.4862
0.3331
0.5148
0.4434
TimesFM-2.5
Independent
0.4645
0.3164
0.5893
0.4818
0.3687
0.5135
0.4632
TiRex-2
Independent
0.4608
0.3201
0.5790
0.4663
0.3678
0.5165
0.4586
Appendix
Table 8: Temporal-path skill on the 94 tasks in fps-bench; higher is better.
Forecast
Copula
VS ↓
No task-specific fitting
Chronos-2
Independent
0.1491
Chronos-2
Comonotonic
0.1648
TimesFM-3
Independent
0.1446
TimesFM-3
Comonotonic
0.1609
Toto-2.0-2.5B
Independent
0.1560
Appendix
Table 9: Contemporaneous cross-variable variogram scores on 35 multi-target tasks. Entries are arithmetic task means; lower is better, with the lowest value in each protocol group bold. Fitted Gaussian references use task-specific calibration.
Forecast
Copula
FPS ↑
No task-specific fitting
Chronos-2
Independent
0.4699
Chronos-2
Comonotonic
0.4434
TimesFM-3
Independent
0.4871
TimesFM-3
Comonotonic
0.4590
Toto-2.0-2.5B
Independent
0.4261
Appendix
Table 10: Effect of fitting a Gaussian temporal copula on FPS over 94 common fev-bench tasks. Each baseline uses the same marginal forecasts across copula completions. Fitted Gaussian rows use task-specific calibration. Bold marks the best value within each protocol group.
sMAPE
Dynamics
Model
H=64
H=128
H=256
H=512
W1
PSD H2
CD err.
Seg. analog
PANDA
19.09
24.42
31.94
41.22
0.588
0.195
0.537
10.03
PathFM-75M
16.94
22.16
28.22
35.37
0.263
0.063
0.286
5.07
PathFM-126M
16.45
21.52
27.43
34.50
0.251
0.061
0.273
4.80
Appendix
Table 11: Model size at ten Euler evaluations per block. Both PathFM sizes follow the protocol in Section F.1 ; lower is better, with the best value in bold.
Figure 14: Accuracy versus sampler budget for both model sizes. Point errors average all 480 source groups; W1 averages all 135 systems and uses 4,096-step sampled continuations. The dotted horizontal line marks PANDA; lower is better.
Model
Sampler
NFE
H=64
H=128
H=256
H=512
PathFM-75M
Euler
1
20.99
26.44
32.97
41.27
Euler
2
17.81
23.56
30.23
37.34
Euler
10
16.94
22.16
28.22
35.37
PDD100k
1
17.62
22.89
29.02
36.31
PDD100k
2
17.17
22.44
28.55
35.69
PathFM-126M
Euler
1
20.14
25.43
31.88
40.14
Appendix
Table 12: Mean source-group sMAPE for the Euler teacher and its student after 100k PDD updates. NFE counts evaluations per 256-step block and sample; bold marks the lowest error within each size.
Figure 15: Full 4,096-step Lorenz and Rössler continuations. All panels use the same initial conditions. Gray curves show the observed continuation; circles mark the first and last forecast states, and shading marks the first 512 steps.
Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.
Time series foundation models (TSFMs) enable cross-domain forecasting, but their development as general-purpose forecasters remains constrained by underexplored training potential and limited architectural versatility. To address these challenges, we introduce Aurora-X, a billion-scale TSFM with a progressive curriculum and a unified architecture. We first use channel-independent pretraining to learn temporal patterns, then introduce cross-variable dependencies, varied context and horizon lengths, and future covariates if available during midtraining. Variable-resolution post-training further enables an adjustable temporal span per token at inference. With fixed model weights, this supports longer histories under a fixed token budget or fewer tokens for the same history, enabling test-time scaling. With a versatile architecture, Aurora-X supports cross-variable modeling, covariate conditioning, and parallel decoding of future patches for probabilistic forecasting. These are supported by a novel pattern-guided mixture-of-experts that expands model capacity through sparse activation and uses shallow patch similarities to constrain deep-layer routing, guiding expert specialization across heterogeneous time series. Furthermore, we propose an implicit quantile network head that predicts arbitrary quantiles to characterize predictive distributions, enhancing probabilistic forecasting flexibility. Comprehensive experiments on GIFT-Eval, TIME, FEV-Bench, TFB, and DAG-Bench demonstrate state-of-the-art forecasting performance against pretrained TSFMs and task-specific supervised models.
Encoder-based time series foundation models (TSFMs) typically rely on deep stacks of independently parameterized Transformer layers, where only the final forecast is supervised and intermediate representations have no explicit predictive role. We introduce FlowTSFM, an encoder architecture that interprets depth as a recurrent transport process: a single Transformer block is iteratively applied with shared parameters, while a quantile-flow objective supervises intermediate states along a prescribed trajectory from a prior distribution toward the final forecast. The objective combines pinball forecasting loss with path-level position matching. With only 38.8M parameters, FlowTSFM achieves competitive performance on GIFT-Eval and TIME, remaining within 1.8-4.6% MASE of stronger baselines while using approximately 3× fewer parameters than a 12-layer Chronos-2 model (119.5M). Beyond accuracy, we introduce CosMean, a scale-free diagnostic measuring whether recurrent updates consistently align toward the final prediction. Under a matched intermediate-state probing protocol, FlowTSFM achieves a CosMean score of 0.919 compared with 0.350 for Chronos-2, suggesting that recurrent parameter sharing combined with path supervision is associated with substantially more structured predictive trajectories at a favorable accuracy-efficiency trade-off.
Bahaeddine Abdessalem, Shifeng Xie, Zehao Xiao +6
Huawei Noah’s Ark Lab, Paris, France · École Polytechnique, France · LIPADE, Université Paris Cité, Paris, France +3
Accurate uncertainty quantification is critical for many time series forecasting applications, and recent studies have increasingly adopted generative models to capture predictive uncertainty. Existing approaches often combine deterministic forecasting with generative modeling to improve predictive performance. However, their joint training is typically unstable because the deterministic model is embedded within the generative process, resulting in deep computational graphs and changing the probability transport path. In this work, we propose DV-Flow, a unified flow-matching formulation that incorporates the deterministic predictor as a parallel velocity component rather than as the endpoint of the transport path. This design enables stable end-to-end optimization while subsuming several existing formulations as special cases, allowing direct comparisons among them. Extensive experiments on six real-world datasets demonstrate that DV-Flow consistently improves probabilistic forecasting performance over strong baselines. Our code is available at https://github.com/tufts-ml/dv-flow-time-series.