Synapse: Adaptive Arbitration of Complementary Expertise in Time Series Foundational Models
Authors: Sarkar Snigdha Sarathi Das
Palash Goyal
Mihir Parmar
Yiwen Song
Long Le
Lesly Miculicich
Jinsung Yoon
Rui Zhang
Hamid Palangi
Tomas Pfister
Venue: Accepted by TMLR ()

Abstract: Pre-trained Time Series Foundational Models (TSFMs) represent a significant advance, capable of forecasting diverse time series with complex characteristics, including varied seasonalities, trends, and long-range dependencies. Despite their primary goal of universal time series forecasting, their efficacy is far from uniform; divergent training protocols and data sources cause individual TSFMs to exhibit highly variable performance across different forecasting tasks, domains, and horizons. Leveraging this complementary expertise by arbitrating existing TSFM outputs presents a compelling strategy, yet this remains a largely unexplored area of research. In this paper, we conduct a thorough examination of how different TSFMs exhibit specialized performance profiles across various forecasting settings, and how we can effectively leverage this behavior in arbitration between different time series models. We specifically analyze how factors such as model selection and forecast horizon distribution can influence the efficacy of arbitration strategies. Based on this analysis, we propose Synapse, a novel arbitration framework for TSFMs. Synapse is designed to dynamically leverage a pool of TSFMs, assign and adjust predictive weights based on their relative, context-dependent performance, and construct a robust forecast distribution by adaptively sampling from the output quantiles of constituent models. Experimental results demonstrate that Synapse consistently outperforms other popular ensembling techniques as well as individual TSFMs, demonstrating Synapse's efficacy in time series forecasting.

Published in Transactions on Machine Learning Research (03/2026)
Synapse: Adaptive Arbitration of Complementary Expertise in Time Series Foundational Models
Sarkar Snigdha Sarathi Das1.2.1, Palash Goyal', Mihir Parmar Yiwen Song!, Long T. Let, Lesly Miculicich', Jinsung Yoon', Rui Zhang", Hamid Palangi and Tomas Pfister Google, 2 Pennsyleaana State University
Corresponding author(s): sf sfd55250psu. edu, (palashgoyal, haw.dppullagggg))oggggggggg
Work done while Sarkor was a Student Researcher at Google.
"oonn last authors
Reviewed on OpenReview: https: // oww.eviee......e..e. net/ net/ forum? id- j3tgggtuSuutt
Abstract
Pre-trained Time Series Foundational Models (TSFMs) represent a significant advance, capable of forecasting diverse time series with complex characteristics, including varied sea- sonalities, trends, and long-range dependencies. Despite their primary goal of universal time series forecasting, their efficacy far from uniform; divergent training protocols and data sources cause individual TSFMs exhibit highly variable performance across different forecasting lasks, domains, and horizons. Leveraging this complementary expertise by ar- bitrating existing TSFM outputs presents compelling strategy, yet this remains largely unexplored area of rescarch. In this paper, we conduct thorough examination of how different TSFMs exhibit. specialized performance profiles across various forecasting settings, and how we can effectively leverage this behavior in arbitration between different time series models. We specifically analyze how factors such as model selection and forecast horizon distribution can influence the efficacy of arbitration strategics. Based on this analysis, we propose SYNAPSE, novel arbitration framework for TSFMs. SYNAPSE: is designed to dy- namically leverage pool of TSFMs, assign and adjust predictive weights based on their relative, context-dependent performance, and construct a robust forecast distribution by adaptively sampling from the output quantiles of constituent models. Experimental results demonstrate that SYNAPSE consistently outperforms other popular ensembling techniques as well as individual TSFMs, demonstrating SYNAPSE'S efficacy time series forecasting.
1 Introduction
Time series forecasting is pivotal task that underpins decision- in high-stakes domains, from manag-
ing energy grids (Zhou et al. 2021 Lai et al. 2018), and supply chains (Mancuso et al. 2021 to navigating financial markets (Godahewa et al. 2021). Historically, the diversity of time series patterns necessitated a specialized approach, compelling practitioners to select different algorithms for different types of time series data. The advent of deep learning, particularly large-scale Transformer- create a universal forccasting solution, pushing the boundarics of stato-of-the-art performance for this task by learning time series foundation models (Das et al. 2024: Graf al., 2025 Woo et al. 2024 Cohen al., 2025 Shi et al. 2024). These models are trained on large corpora of time series data to learn the ability to identify specific patterns in mumerical data. At convergence, the model weights capture the average dynamics of the training distribution.
architectures, have sought to
However, this monolithic approach harbors fundamental flaw: forces the model into a representational compromise. Real-world time scries are inherently non-stationary, composod of complex mixture of competing patterns-stable scasonalities, long-term trends, and abrupt structural breaks. A single, static model averages these conflicting signals, rendering it brittle and unprepared for sudden regime shifts, where

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Domain
Horizon
Modal (Oracle)
Entropy (Bits)
Modal Share
Switch
Ecorffin Energy
Frog. %
Moirai2
Sundial
Healthcare
Econ/Fin
Short
Sundial
2.541
0.251
14.47%
Nature
Long
Moirai2
2.432
0.281
45.31%
Transport WebrCloudOps
Energy
Medium
Sundial
2.421
0.288
41.86%
Short
Sundial
2.467
0.278
43.53%
Healthcare
Short
Sundial
2.531
0.237
41.28%
Long
Sundial
2.505
0.283
29.36%
Moirai-L
10%
20%
30%
18to
Nature
Medium
Sundial
2.497
0.285
27.79%
Short
Moirui-L
2.544
0.213
40.82%
Sales
Short
Sundial
2.550
0.242
47.27%
Long
Moirai-L
2.494
0.278
51.75%
Transport
Medium
Moirai-I L
2.529
0.254
51.08%
Short
Sundial
2.550
0.236
58.45%
Loug
Moirai2
2.562
0.225
49.31%
Web/
MoiraiBB
Moiral-S
CloudOps
Medium
Moira12
2.556
0.223
50.09%
Short
Toto
2.509
0.240
49.67%
Figure 1: (Left) Oracle arbitrator's model selection frequency across seven different domains of G-Fl--vvll
(Aksu et al. 2024). These values represent the percentage of timestamps each model was selected by Oracle as the optimal predictor across different domains. (Right) Detailed Oracle analysis by domain and forecast horizon. The Modal (Oracle) column identifies the Time Series Foundation Model (TSFM) most frequently selected as the optimal predictor by the Oracle, while Modal Share quantifies the proportion of time that model held that majority. Selection Entropy (where 2.58 bits denotes uniform distribution across 6 models) provides quantitative measure of selection diversity, and Switch Frequency represents the average percentage of timesteps where the Oracle's preferred model changes. These results demonstrate that model expertise is broadly distributed across the pool; the high entropy values and low modal shares provide a strong empirical justification for our arbitration approach, as a static selection would be consistently suboptimal.
performance can degrade catastrophically. While simple static ensembles are common mitigation strategy, they merely average these prready-comprmmiset representations. While this need for dynamic specialization is conceptually related to Mixture of Experts (MoF) frameworks (Shazeer et al. 2017 Zhou et al. 2022 Liu
ct al. 2024), , where a gating network routes inputs to specialized sub-models, MoE architectures are typically trained end-to-end, jointly optimizing the gale and their constiluent experts. HHanng et al. (2025) explores framing this problem as multi-specialistt cooperation by decomposing forecasting into sub-tasks handled by specialized agents. Our work addresses a diffcrent and, in the context of TSFMs, more pressing challenge: performing post-hoc arbitration between pre-erisling, monolithic, and fered-weight foundational models.
To validate this paradigm, we first quantify its empirical upper bound using a temporal Oracle that, at each timestamp, selects the most accurate prediction from diverse pool of expert models. Examining the selection frequency of optimal model prediction at timestamp level in Figure (left), we observe that optimal model can vary widely across different timestamps, and their usage percentages also vary widely across different domains. We find that the Oracle frequently relies on weaker, specialized models that excel only in specific regions. Furthermore, the number of model switches required for optimal prediction is substantial (Figure I right), proving static selection is insufficient. This evidence highlights the fundamental constraints of traditional ensembles. Such approaches apply a fixed aggregation rulc-like centering predictions or taking the median of quantiles (Garza & Rosillo, 2025), which overlooks informative output distributions, yields suboptimal performance (Table L), and is fundamentally unable to adapt to the highly dynamic, timestamp- to-timestamp nature of model performance.
Informed by these observations, we introduce SYNAPSE, novel TSFM arbitration framework for lime series forecasting. SYNAPSE keeps a pool of TSFMs, and dynamically arbitrates between them at timestamp level granularity leverage the predictions from the best forecasting models, while de-prioritizing the less accurate models for a specific timestamp. For each timestamp, SYNAPSE merges the inverse quantile predictions from its constituent models to give the final output distribution. To dynamically adjust the arbitration
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timestamp level, SYNAPSE maintains moving window, which it leverages for selecting and adjusting the weights for arbitration.
Our experimental results, conducted on the extensive GIFT-eval benchmark of 23 datasets, provide strong evidence for the superiority of the dynamic arbitration paradigm. SYNAPSE achieves new state-of- shate-of-the-art performance, with performance gains consistently amplifying over longer forecast. horizons. The strength of SYNAPSE is highlighted by its ability to consistently arbitrate a committee of specialists outperform any single member, demonstrating its potential to sometimes surpass single, monolithic stronger model. This result proves that the adaptive arbitration mechanism itself, is the primary driver of performance. Our primary contributions are:
We identify the "representational compromise" inherent in monolithically trained models as core weakness for non-stationary time series forecasting. Subsequently, we built "Oracle' arbitrator, that optimally selects the best model forecast at each timestamp to set empirical performance upper bound in time scries forccasting with current models.
We re-frame the forecasting task and define a dynamic arbitration problem between TSFMs. We then propose SYNAPSE, novel arbitration framework that dynamically arbitrates the influence of constiluent models at each time step.
We show that SYNAPSE achieves state-of-the-art performance, with gains that amplify over longer horizons, where baseline ensemble approaches falls apart due static mixture of ensembles. We directly link this to its dynamic weight-shifting mechanism.
Finally, we demonstrate that the arbitration mechanism is the key driver of performance, enabling SYNAPSE to effectively synthesize complementary expertise and consistently improve upon the ca- pabilities of its individual constituent models.
Our work establishes that adaptive arbitration is a powerful and necessary paradigm for building the next generation of robust, real-world time series forecasting systems.
2 Related Works
The field of time series forecasting has evolved through several distinct paradigms, starting with statistical methods, advancing to task-specific deep learning models, and most recently loveraging large-scale founda- tional models. Our work is situated at the intersection of this latest paradigı and the long-standing practice of model ensembling, adapted for the unique challenges and opportunities presented by modern TSFMs.
Statistical and Deep Learning Methods Time series analysis has historically relied on statistical methods like the ARIMA family (Box al. 2015 Box & Jenkins, 1976), ETS (Hyndman et al. 2008).
and additive regression Taylor & Letham 2018). While robust baselines, they often struggle with complex non- -ineinnar dependencies. The advent of deep learning introduced automated pattern learning, evolving from RNNs (Hochreiter & Schmidhuber 1997] and probabilistic models (Salinas et al. 2020) to specialized architectures like N-BEATS (Oreshkin et al. 2019 and TimesNet (Wu et al.) 2023). Recently, the field has debated the efficacy of Transformers (Zhou et al. 2021 Wu al.) 2021) versus simpler MLP-based models (Chen et al. 2023 Zeng et al., 2023), with innovations like PatchTST (Nie al. 2023) bridging the gap. Despite their power, these deep learning models are typically trained for specific tasks and datasets, requiring significant data and computational resources for each new application.
Time Series Foundational Models The latest paradigm shift in forecasting mirrors the revolution in natural language processing with large language models: the rise of pre-trained Time Series Foundational Models (TSFMs). These models, trained on massive and diverse datasets, perform zero-shot forecasting on unscen time scries with remarkable accuracy. A prominent approach treats forccasting as language modeling problem, where continuous time series values are tokenized into a discrete vocabulary. Models like Chronos (Ansari et al. 2024) and TimeGPT (Garza et al. 2023) exemplify this, using standard large language
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model (LLM) architectures to predict the next token. Building on this, Chronos-2 Ansari et al. 2025
evolves the architecture from univariate predictor to "universal" forecaster, leveraging group attention mechanism and synthetic training data to natively handle multivariate dependencies and external covariates. A contrasting approach involves direct regression on contimuous values. TimesFM (Das et al., 2024] is a
leading example, employing a decoder-only architeclure pre-trained on 100 billion time points for extreme long-horizon forecasting. Other models explore novel pre-training strategics and architectures, including generative state-space models like Flowstate (Graf et al.) 2025). The Moirai family (Woo al. 2024) particularly illustrates rapid architectural evolution: starting
masked encoder with multi-pated layers to handle diverse frequencies, it evolved into Moirai-MoF. (Liu et al. 2024) utilizing sparse Mixture-of-Fxperte Most recently, Moirai 2.0 (Liu et al. 2025a) shifts towards a more streamlined decoder-only architecture, utilizing quantile loss to achieve superior probabilistic accuracy and inference speed with significantly fewer parameters.. The core promise of TSFMs a universal forecasting solution, yet as our Oracle analysis demonstrates, no single model universally dominates. Instead, they exhibit complementary expertise, making them prime candidates for advanced arbitration.
3
Preliminaries and Insights
3.1 Time Series Forecasting
The fundamental task of time series forecasting is to predict future values of sequence based on its observed history. Therefore, this problem can be framed as a mapping from historical context data to future horizon prediction.
Formally, let X = (I1, "e..., xT) represent a iaivaariate/multivaariii time series of variables observed over context window of length L. The primary objective generate a forecast for the subsequent T timesteps, known the forecast horizon, denoted X Modern forecasting approaches move beyond simple point estimates to generate full predictive distribution, P(X,+1 (1 which captures the inherent uncertainty through forccast quantiles.
The quality of this forecast is then evaluated against the ground truth, Y = X using metries that assess both probabilistic accuracy and point-wise error. Commonly used metrics for these are Continuous Ranked Probability Score (CRPS) and Mean Absolute Scaled Error (MASE) respectively. Details about these metrics can be found in Appendix B
3.2 Arbitration of Time Series Forecasting Models Problem Formulation
Let M = (M,, Myy } be a pool of N distinct Time Series Foundational Models (TSFMs). For a given input XI.r, each model M; E M generates its own predictive distribution (x(x+1144|XX X for the forecast horizon.
Our proposed arbitration setting aims to synthesize these individual forecasts into a single, superior predictive output. Instead of simple aggregation, we define dynamic arbitration process where the final forecast is a predictive distribution, constructed by an arbitration function, A, that operates on constituent models' output distributions.
The arbitrated predictive distribution for timestep t, represented by its set of quantiles Qarb, is given by:
Qua Al((wi,, Qud M, E M})
where:
Q. represents the predictive quantile distribution from model Mj for timestep
wiy is a dynamic, non-negative weight assigned to model M, at timestep t, reflecting its expected performance.
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This framework can be specialized to represent traditional ensembles. For example, simple mean ensemble of point forecasts can be formulated when weights are uniform (Wi,t 1/N), and A is defined as the averaging function over the median of each input distribution Qi.--
For the scope of this work, we limit our experiments to a pool of six highly pop- ular Time Series Foundational Models: Sundial (Liu et al. 2025b), Toto (Co- hen et al. 2024), Moirai2, Moirai-large, Moirai-base, and Moirai-small. (Woo
Hodel
Moiral-malll
0.946
Moiral base
0..011
Moirai-arrge
0.875
Toto
Sundial
Moiral2
et al. 2024
0.728
Oracle
0.650
3.3 Oracle Selection Arbitrator
Scores
106110
0.599
0.559
0.518
0.517
0.516
To establish an empirical performance expectation for selective arbitration strategy, we define an Dracle-Based Selection Arbilralor. This conceptual model operates with perfect foresight, possessing knowledge of the ground truth outcomes. At each timestep in the forecast horizon, the Oracle se- lects the single, optimal TSFM from the pool M, that is, the model whose predictive distribution is closest to the actual value.
0.356
MASE
CRPS
Figure 2: GIFT-Eval Performance comparison (MASE CRPS) of the Oracle Arbitrator against individual constituent TSFMs. Ora- cle selector can outperform all other constituent models by large margin, demonstrating the efficacy of a strong arbitrator.
Formally, the predictive distribution selected by the Oracle at timestep t is Opracle, defined as:
Quracle = Q+ where i"= min CRPS(Q...V++.)
()......N
Here, Vr+t is the ground truth value at the future timestep.
The performance of this Oracle thus represents the lowest possible error achievable the best model could be identified at every single step. The significant potential of dynamic arbitration is highlighted in Figure which compares the performance of the Oracle against the individual TSFMs in our pool.
Furthermore, analyzing the Oracle's selection percentages reveals an important insight: no single model dominates across all timestamps. As illustrated in Fi the selection frequency of each TSFM varies considerably across different domains. Notably, models that exhibit weaker overall performance (e.g., Moirai- small) are still frequently chosen as the optimal predictor for specific timestamps, highlighting the context- dependent nature of forecasting expertise and reinforcing the need for an adaptive arbitration mechanism.
Oracle Analysis and Selection Diversity Beyond establishing an upper bound, the Oracle provides deep insights into the transient nature of model expertise. In Figure [ we analyze the Oracle's choices across GIFT-Eval domains and horizons using four key metrics. If we let p; represent the empirical selection probability of model M, over the forecast horizon. The Modal (Oracle) choice identifies the model that holds the lead most frequently for being chosen (i" arg max; Pr), while the Modal Share (p:+) quantifies the percentage of timesteps the most frequent model held the lead. To measure ion diversity across the pool, we compute the Selection Entropy:
N pi log, Ps
H =
Σ
(1)
In our experiments, N 6 for the pool of six TSFMs, therefore the maximum theoretical entropy for this pool is (6) R 2.585 bits, representing perfectly uniform distribution of expertise.
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TSFM Inferences and Weight Initialization
2 Arbitration via Predictive Sampling
Model
M
W
Predictive
Sampling Arbitrator
weighited sempling
Model 2
W2
weightte sampling
Model 3
Weight
Initialization
Wy
weighted sampling
Historical Context
Input
Model
W
weightted sampling
Dynamic (M,- O2
Weight Adaptation
Dynamic Weight Adaptation via Forward Simulation
Figure 3: Overview of Synapse. It takes a Historical Conteat Input (left), which is then fed into a pool of N diverse Time Series Foundational Models (TSFMs). Each model produces its probabilistic forecast. These individual forecasts are then fed into the core arbitration mechanism. At each timestep 1, a set of Dynamic Weights {wi..... ] is applied to the corresponding model forecasts, which is leveraged in the predictive sampling and in subsequent construction of the final Arbitrated Forecast Distribution for that timestep. This entire process governed by a Dynamic Weight Adaptation via Forward Simulation loop: the arbitrator's own forecast from previous step used as a simulated observation, which is then used update a rolling performance window, which recalculates the CRPS for all models to determine the dynamic weights {wi,++1} for the next timestep.
As shown in the table in Figure (right), the Selection Entropy values consistently exceed 2.4 bits, approaching the theoretical maximum. This demonstrates that Oracle optimality is broadly distributed across the pool rather than being concentrated on single dominant model with rare "flips." Furthermore, the Modal Share remains low (typically 20%-28%), meaning that even the best model in domain is suboptimal for over 70% of the horizon. Finally, the high Switch Frequency (often > 50%) confirms that model leadership highly transient across timesteps. This persistent and high-frequency switching underscores that no single model optimal for an entire forecast window. In fact, as we demonstrate Appendix C even the best domain-specific static model substantially underperforms a dynamic arbitration strategy. Together, these metrics demonstrate a fundamental "representational compromise" in monolithic TSFMs and justify the necessity of the dynamic arbitration provided by SYNAPSE.
4 Synapse: Dynamic Arbitration
SYNAPSE is designed to close the performance gap to the Oracle selector by dynamically adjusting its strategy at each step of the forecast horizon. As demonstrated in Figure 3. SYNAPSE's core mechanism consists of two key components: (1) a weighted arbitration function based on predictive sampling. (2) a dynamic weighting scheme based on a rolling performance window that performs forward simulation mechanism allowing to adapt over the forecast. horizon. For this implementation, utilize the full set of models at each step.
Built on the definition of general arbitration framework (Section 3.2

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4.1 Arbitration via Predictive Sampling
We design the arbitration function, A, to create nou-parametric mixture of the underlying distributions they represent. We achieve it through a weighted predictive sampling process, which guarantees that the resulting arbitrated quantiles, Q are probabilisticall valid (i.c., monotonic) by construction, climinating the need for post-processing.
Sample Gencration For cach model M,, we generate n, random samples from its predictive distribution. In this regard, we apply inverse transform sampling method to the model's estimated quantile function. The final arbitrated quantile for given level a, denoted (arh is then calculated as the empirical a-th quantile of the pooled set of all generated samples. We formally express this entire process as:
UEA -00)), TM where n, [Nvotall Wa,e]
Latt
Quantile
.a)
(2)
Here n, is the number of samples allocated to model M,, determined by its performance weight wi,e- Here, ()) is the inverse Cumulative Distribution Function (CDF), estimated from the discrete quantile predic- tions of model M, at time t. The continuous function Pi.1 is practically approximated by fitting cubic spline to the discrete quantile points for probabilities within [0.1, while stable linear extrapolation is used for the tails of the distribution.
This sampling process yields a pooled set of samples, P. - (samples from M.). The final arbilrated predictive distribution, Qu+b, is the set of empirical quantiles calculated from this collection P.. This method ensures that the final distribution faithfully represents a true mixture of the constituent predictive distribu- tions.
4.2 Dynamic Weight Adaptation via Forward Simulation
Although Equation 2 presents a compelling strategy of dynamically adjusting the weight. of model predictions at different timesteps to account for the changing performance trends of different models, the primary challenge here is to calculate the weight in the absence of ground truth value for the prediction horizon. To address this, we employ a forward simulation to adapt the model weights, (w,,e), at each step. The weights are determined by each model's recent performance, mcasured by its CRPS score over rolling historical window of length W. The primary weighting rule is inverse error weighting (wa, x (CRPSS..-..-ll... with robust softmax fallback to handle the numerical instability of near-zero scores. This entire adaptation mechanism operates as a foedback loop:
Step t Prediction: At timestep in the horizon, the arbitrator calculates the final arbitrated quantiles, Qarb, using the current set of weights {war].
2. Simulated Ground Truth: It then defines a simulated ground truth, yi for that timestep by taking the median of its own prediction: y: = median(Q.rh)
3. Performance Window Update: This simulated observation is used to update the rolling perfor- mance window. The oldest data point in the window is discarded, and a new record, consisting of the simulated ground truth (yg and the original predictions from all models for that step (+Q..)), is added.
4. Weight Update: The CRPS scores and, consequently, the weights (wi,e-}} for the neat timestep, L ++1 1, are then recalculated based on this newly updated performance window.
This process repeats for every step in the forecast horizon, allowing the arbitrator to dynamically learn and adapt its strategy basod on a simulation of its own ongoing performance. Intuitively, this allows SYNAPSE to create a self-reinforcing feedback loop: models whose predictions align well with the aggregated mixture median (the simulated ground truth) will achieve lower CRPS score in the subsequent step and thus receive
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Algorithm SYNAPSE
Input: Pool of TSFMs M (M....., My), Forecast horizon T, Initial performance window Winit: Total samples Notall Quantile levels {ax}
2: Output: Final arbitrated forecast distributions { It=1
3: W t Winit
Initialize rolling performance window Initialize output container
(Qarb)7 I==1 -- 0
for t t 1 to T do
// Dynamic Weight Calculation for i + to N do
7:
8:
8; < AverageCRPS(M, W) if min(fs;)) then
D Calculate score for each model fallback
10:
(wi.t} Normalin((Softmax((>>> 8;)))
11: else
12:
{wi,e} < Normalize((mverreerro(((((]
13:
// Arbitration via Predictive Sampling
14:
P. <-
15:
for to N do
16:
ni [Ntotal -W,rl
Calculate number of samples take from each model
Fi, FitSplinee (Q.))
17:
Pit (ps) Pj (O, =
18:
19:
P. + PiUPi
Qarb + Quantile(P;, {ax})
20:
21:
// Forward Simulaatio and Window Update alm +- median( (Qae))
22:
23:
Onew t (2/ {
24:
W + (W fooldest }) U {Ouew
b Update rolling window
25: return 1T
higher weight. On the other hand, models that consistently diverge from the emerging consensus are penalized with lower weights, effectively reducing their influence over time. While this forward simulation relies on the aggregated consensus as a proxy for ground truth, our ablation study (detailed in Table 3) demonstrates this a key driver of SYNAPSE'S success, as the SYNAPSE with dynamicity substantially outperforms static weights.
5 Experiments
In this section, empirically validate the effectiveness of SYNAPSE. Our experimental setup is grounded in the framework established in our preliminaries, focusing a pool of highly popular TSFMs. We conduct series of experiments to evaluate SYNAPSE's overall performance, compare it against strong ensemble baselines. Furthermore, we perform a rigorous ablation study to dissect the components contributing to the improvements, followed by a performance analysis across different scenarios.
5.1 Experiment Setup
Benchmark Dataset To demonstrate the effectiveness of SYNAPSE, we evaluate and compare its perfor- mance in GIFT-Eval comprehensive benchmark Aksu al. 2024 This benchmark consists of 23 datasets

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Table 1: Overall performance comparison on the GIFT-Eval benchmark, broken down by forecast horizon.
The best model performance in each column in bold, and the second best is underlined.
Model
Long Horizon
Medium Horizon
Short Horizon
Overall
CRPS
MASE
CRPS
MASE
CRPS
MASE
CRPS
MASE
Moirai2
0.513
0.789
0.519
0.759
0.517
0.695
0.516
0.728
Toto
0.496
0.812
0.499
0.772
0.533
0.720
0.517
0.750
Sundial
0.510
0.785
0.526
0.764
0.592
0.732
0.559
0.750
Moirai-small
0.626
1.035
0.636
1.021
0.666
0.888
0.650
0.946
Moirai-base
0.611
1.029
0.632
1.017
0.601
0.818
0.610
0.901
Moirai-large
0.598
0.985
0.605
0.961
0.597
0.807
0.599
0.875
Mean Ensemble
0.502
0.846
0.514
0.82
0.535
0.727
0.523
0.771
Median Ensemble
0.481
0.793
0.493
0.767
0.523
0.707
0.517
0.762
Synapse 0.464
0.756
0.472
0.733
0.517
0.7
0.496
0.719
Oracle
0.329
0.530
0.328
0.512
0.379
0.517
0.356
0.518
across 7 different domains having over 144,000 time series, featuring 97 unique combinations of dataset, fre- quency, and length. Table T shows the overall performance comparison of different methods GIFT-Eval.
Baselines As denoted in Section 3.2, we primarily consider Sundial [Liu et al. 2025b), Toto (Cohen et al. 2024), Moirai2, Moirai-large, Moirai-base, and Moirai-small (Woo et al. 2024) as the foundational models. We compare SYNAPSE performance to that of these constituent models; additionally we compare against recent strong baseline median-ensemble (Garza & Rosillo 2025) that takes median values of multiple models at each quantile, its variant mean-ensemble, and standard point forecast means as our baselines.
5.2 Overall Performance
Table presents the CRPS and MASE scores for each of the six constituent TSFMs, median ensemble, and mean ensemble compared against the performance of SYNAPSE. The results clearly demonstrate that SYNAPSE significantly outperforms every individual model in the pool, well the ensemble baselines. Notably, it achieves CRPS of 0.496, which is marked improvement over the best-performing individual model, Moirai2 (0.516). This showcases the ability of our dynamic arbitration mechanism to successfully synthesize the varied expertise of the foundational models into single, more accurate predictive distribution.
5.3 Comparison with Ensemble Baselines
For comparison with ensemble techniques, we choose median-ensemble (Garza & Rosillo 2025). recent strong ensemble baseline that leverages medians quantiles for time series prediction. While the Median Ensemble (MASE 0.762) offers slight improvement over some individual models, it fails to match the per- formance of the best-performing constituent TSFM (Moirai2, MASE 0.728). In contrast, SYNAPSE (MASE 0.719) substantially outperforms both the Median Ensemble and the best constituent individual model, confirming that its dynamic and adaptive nature provides a distinct advantage over simple aggregation strategics.
5.4 Performance Across Forecast Horizons
Table [ also details the performance trends across different prediction horizons. To visualize these trends more clearly, we examine the line plots in Figure 4 The plots illustrate that as we move from short horizon to medium and longer horizon, the efficacy of SYNAPSE arbitration framework also grows. More specifically, for short-horizon forccasts, SYNAPSE'S performance individual model for that range, Moirai2, However, as the task difficulty increases for medium and long- horizon predictions, SYNAPSE. establishes distinct performance advantage. It begins to more effectively
competitive with the best-performing

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CRPS Performance
MASE Performance
0.075
1.55
MAk5
0.000
CRPS (Lower Bettel)
0.625
MASE Score (Lower is Better)
Sundial
7090
0.35
Mataiamall
Maina base
0.000
0.57
Synapse
02:00
0.525
0.500
0.475
0.70
Medikm
Meduu
Forecast
Forecast Horizon
Figure 4: Performance of all models across different forecast. horizons (Short, Medium, Long). SYNAPSE consistently shows strong and often superior performance for both CRPS and MASE, particularly the forecast horizon increases.
leverage the complementary strengths of different models in the pool, widening the performance gap between itself and any single TSFM. In contrast, the Median Ensemble struggles to adapt, often exhibiting higher MASE score than several of its constituent models at longer horizons, which indicates failure to effectively aggregate their predictive power.
5.5 Model Selection Accuracy Comparison
While the ultimate goal of an ensemble is to produce the most accurate final forecast, it is also insightful to evaluate the effectiveness of the underlying arbitration mechanism itself. We define Model Selection Accuracy as a metric that quantifies how well the weighting of SYNAPSE aligns with the choices made by the Oracle Arbitrator (defined in Section 3.2) at each timestep. This metric assesses whether the arbitration correctly identifies the truly best-performing model (or ranks it highly) based on the information available to it. For comparison against primary baseline median-ensembll (Garza & Rosillo 2025), since it does not do explicit weighting, infer its "implicit choice' by ranking the constituent models based the proximity of their median forecast to the final ensemble median forecast. Models whose median predictions are closer to the ensemble's 's final output median are given better rank.
Table 2: Top-k Model Selection Accuracy Comparison of SYNAPSE and Quantile Median Ensemble. Measures the percentage of timesteps where the Oracle's best model was ranked within the top ranking by the corresponding method.
Method
Top-1 (%)
Top-2 (%)
Top-3 (%)
Top-4 (%)
Top-5 (%)
Top-6 (%)
Median Ensemble 10.73
21.79
35.43
53.36
74.25
100
SYNAPSE
23.52
41.57
56.95
70.18
82.87
100
Table 2 presents the Top-k Model Selection Accuracy for both the SYNAPSE arbitrator and the Median Ensemble baseline, averaged across all datasets and forecast horizons evaluated. The results demonstrate that the dynamic weighting mechanism employed by SYNAPSE is significantly more effective at identifying the best-performing model at each timestep compared the implicit ranking of the Median Ensemble, effectively showing the elficacy of SYNAPSE in closing the gap with Oracle.
5.6 Impact of Arbitration and Dynamicity
To deconstruct the sources of SYNAPSE'S performance gains, we conduct an ablation study isolate the contributions of its two primary components: (1) Arbitration via Predictive Sampling and (2) Dynamic
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Weight Adaptation. We compare three variants: (1) Median Ensemble, the baseline following Garza & Rosillo (0225) using no components from our proposed techniques; (2) Synapse (- Dynamicity), which uses our predictive sampling arbitration but with static, uniform weights (removing dynamic adaptation); and (3) Synapse, the full proposed model.
The results Table Baeeeeelaatt clearly shows that shifting from a Median Ensemble to SYNAPSE with static weights ('- Dynamicity") yields a notable performance improvement (MASE drops from 0.762 to 0.738), demonstrating the inherent value of the predictive sampling method for creating superior mixture distribution. The subsequent addition of the dynamic weight adaptation via forward simulation provides another substantial boost, further reducing the MASE to 0.719. This two-step improvement validates the architectural choices of SYNAPSE, proving that both the intelligent mixture strategy and the adaptive weighting scheme are crucial to its success.
Table 3: Domain-wise performance ablation for SYNAPSE, detailing the incremental gains from its core components. We compare the baseline (Median Ensemble), SYNAPSE without dynamic weighting, and the full SYNAPSE. The best-performing model for each metric within each domain is highlighted in bold.
MASE
CRPS
Domain
Lumpiness
Median Ensemble
Synapse
Synapse (Full)
Median Ensemble
Synapse (- (-Dynamicity)
Synapse (Full)
(- (-Dynamicity)
Econ/Fin
0.14
0.801
0.802
0.784
0.738
0.745
0.737
Energy
0.24
0.870
0.845
0.825
0.617
0.603
0.592
Healthcare
0.07
0.632
0.633
0.620
0.491
0.503
0.471
Nature
1.09
0.722
0.709
0.711
0.342
0.337
0.345
Sales
1.54
0.688
0.691
0.690
0.416
0.417
0.416
Transport
0.39
0.601
0.595
0.588
0.450
0.445
0.440
Wie//ClodOOps
1.47
0.808
0.737
0.690
0.561
0.534
0.497
Overall
0.762
0.738
0.719
0.517
0.507
0.496
5.7 Domain Wise Performance Ablation
To further understand the behavior our framework, we present a performance breakdown across the seven distinct domains within the GTFT-Fval benchmark. Table 3 details the performance of SYNAPSE against Quantile median ensemble (Garza & Rosillo 2025) in each domain. The results show that SYNAPSE generally outperforms the static median-cnsenie ensemble baseline. Please note that datasets in "Sales" domain (e.g. Restaurants, Hierarchical Sales, Car Parts) has substantially shorter forecast horizon (8-30) which does not provide sufficient runway for SYNAPSE to mcaningfully adapt the weights, Consoquently, we see comparable results to that of Median Ensemble.
To further analyze this in the context of domain-specific data characteristics, conducting a correlation anal- ysis between the performance difference (Modian Ensemble MASE SYNAPSE MASE) and the lumpiness of each domain, found positive correlation score of 0.32 suggesting that the dynamic arbitration mecha- nism be beneficial navigating the inconsistent. data patterns found in highly variable, non-stationary environments.
5.8 Synapse: Performance Scaling with Models
The primary strength of SYNAPSE is its ability to synthesize the complementary expertise of multiple TSFMs. Therefore, it is imperalive to understand how its performance scales as the pool of available models expands over time. We simulate this evolution by establishing a fixed, chronological order for the constituent models- Moirai-small, Moirai-base, Moirai-large, Sundial, Toto, and Moirai-2 and then constructing ensembles of incrementally increasing size. Starting with the first two models, we cumulatively add the next model in the sequence to form ensembles of three, four, five, and ultimately all six models.
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At each step, compare the performance of SYNAPSE against the single best-performing model within that specific pool, based on its CRPS. The results, as shown in the Figure 5 reveal a crucial characteristic of our method. SYNAPSE consistently outperforms the best individual model available in the pool, regardless of the ensemble size. This demonstrates that the synergistic combination of expertise within SYNAPSE is more powerful than relying on any single, lop-performing TSFM.
This robust scaling behavior is a strong indicator of the future-proof nature of SYNAPSE. As newer and potentially more powerful TSFMs emerge, this trend sug- gests that SYNAPSE will not only improve its absolute performance but might also continue to maintain an advantage over the best monolithic model.
individual CXP
0.56
0.4
6 Conclusion
2.7
In this work, we identified the "represen- tational compromise" in monolithic Time Series Foundational Models key bar- rier to robust forecasting, particularly for non-stationary data. We established an empirical performance ceiling by de- veloping an Oracle selector, which high- lighted the significant untapped potential in leveraging the complementary, time- varying expertise of existing TSFMs.
0.50
Number Nodels
Figure 5: Performance comparison between SYNAPSE and the best individual model in the pool. As the number of models increases, SYNAPSE consistently achieves a lower (better) CRPS, demonstrating its effective synthesis of model expertise.
To harness this potential and close the gap to Oracle, we introduced SYNAPSE, a novel dynamic arbitration framework.
By employing a forward-simulation mechanism to dynamically adapt its weights and a predictive sampling strategy to construct a probabilistccall sound forecast, SYNAPSE effectively navigates the strengths and weaknesses of its constituent models at each timestep. Our extensive experiments demonstrate that SyNAPSE achieves state-of-the-art performance, significantly outperforming both individual state-of-the-art models and traditional ensemble basclines. We showed that its performance advantage amplifics over longer, more challenging forecast horizons where static ensembles often falter.
Critically, we demonstrated that an arbitrated mix of weaker models, when guided by SYNAPSE, can de- cisively outperform stronger, single monolithic TSFM. This finding suggests a paradigm shift in time series forecasting: the pursuit of a single, universal model may be less fruitful than developing sophisticated methods for dynamically arbitrating ecosystem of specialized experts.
Limitations
While SYNAPSE demonstrates strong performance, it's primary limitation lies in its forward simulation. The process leverages the arbitrator's 's own forecast "simulated ground truth" to update model weights. This self-referential loop risks confirmation bias, potentially favoring consensus models over correct outliers. This is particularly probable in cases, where none of the models can provide very meaningful forecasts resulting
in SYNAPSE often getting lower performance compared to that of some constituent models. Figure 7 (h) and
(i) shows examples where SYNAPSE slighly underperforms constituent models. Future work may focus on breaking this loop, for instance by introducing exploration techniques to ensure that correct, non-consensus models can still gain influence.
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Acknowledgments
We would like to thank Rui Meng, Vishy Tirumalashetty and Burak Gokturk for their valuable discussions, leedback, and guidance throughout this project.
References
Taha Aksu, Gerald Woo, Juncheng Liu, Xu Liu, Chenghao Liu, Silvio Savarese, Caiming Xiong, and Doyen Sahoo. Gift-eval: A benchmark for general time scries forecasting model evaluation. ar preprint arXiv:2410.106: 2024.
Abdul Fatir Ansari, Lorenzo Stella, Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Olek- sandr Shchur, Syama Sundar Rangapuram, Sebastian Pineda Arango, Shubham Kapoor, et al. Chronos: Learning the language of time series. arXiv preprint orXiv:2403. 07815, 2024.
Abdul Fatir Ansari, Oleksandr Shehur, Jaris Küken, Andreas Auer, Boran Han, Pedro Mercado, Syama Sun- dar Rangapuram, Huibin Shen, Lorenzo Stella, Xiyuan Zhang, et al. Chronos-2 2: From univariate to universal forccasting. arXiv preprint arxiv:2510. 15821, 2025.
George Box and GM Jenkins, Analysis: Forecasting and control. San francisco, 1976.
George EP Box, Gwilym M Jenkins, Gregory C Reinsel, and Greta M Tjjug. Time Series Analysis: Fore- casting and Control. John Wiley & Sons, 2015.
Si-An Chen, Chun-Liang Chen, Zsolt Wang, Kuan-Ting Tung, Ching-T Ho, Chien-Chun Lee, and Yu- Chiang Frank Wang. Tsmixer: An all-mlp architecture for time series forecasting. arXiv preprint arXiv:2303.0605 2023.
Ben Cohen, Emaad Khwaja, Kan Wang, Charles Masson, Elise Ramé, Youssef Doubli, and Othmane Abou- Amal. Toto: Time series optimized transformer for observability. arXiv preprint arXiv:2407.07874, 2024.
Ben Cohen, Emaad Khwaja, Youssef Doubli, Salahidine Lemaachi, Chris Lettieri, Charles Masson, Hugo Miccinilli, Flise Ramé, Qiqi Ren, Afshin Rostamizadeh, Jean Ogier du Terrail, Anna-Monica Toon, Kan Wang, Stephan Xie, Zongzhe Xu, Viktoriya Zhukova, David Asker, Ameet Talwalkar, and Othmane Abou- Amal. This time is different: An observability perspective on time series foundation models, 2025. URL
https://arxii: http//aiiv.orrgaaa//aa/aaaaaa wrgg/ba///05....1/
Abhimanyu Das, Weihao Kong, Rajat Sen, and Yichen Zhou. A decoder-only foundation model for time- series forecasting, 2024. URL //axxi/ http:///aaw....o///////1000000. ort//bb///31..1
Azul Garza and Reneé Rosillo. Timecopilot. arXiv preprint arXiv:2509.008: 00616, 2025.
Azul Garza, Cristian Challu, and Max Mergenthaler-Canneee. Timegpt-1. arXiv preprinl arXiv:2310.03588. 2023.
Rakshitha Godahewa, Christoph Bergmeir, Geoffrey Webb, Rob J Hyndman, and Pablo Montero-Manso. Monash time series forecasting archive. arXiv preprint arXic:2105.0666 2021.
Lars Graf, Thomas Ortner, Stanisf. Wofąniak, Angeliki Pantazi, et al. Flowstate: Sampling rate invariant time series forecasting. arXiv preprint Xiv: 052287, 2025.
Sepp Hochreiter and Jürgen Schnidhuber. Long short-term memory. Neural computation, 1735 1997.
Qihe Huang, Zhengyang Zhou, Yangze Li, Kuo Yang, Binwu Wang, and Yang Wang. Many minds, one goal: Time series forecasting via sub-task specialization and inter-agent cooperation. In The Thirty-ninth Anaual Conference on Newral Information Processing Systems, 2025.
Rob Hyndman, Anne B Kochler, J Keith Ord, and Ralph D Snyder. Forecasting with Erponential Smooth- ing: The State Space Approach. Springer Science & Business Media, 2008.
13

---

Published in Transactions on Machine Learning Research (03/2026)
Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long-and short-term temporal patterns with deep neural networks. In The 41st international ACM STGIR conference on research g development in information retrieval, pp. 95 104, 2018.
Chenghao Liu, Taha Aksu, Juncheng Liu, Xu Liu, Hanshu Yan, Quang Pham, Doyen Sahoo, Caiming Xiong, Silvio Savarese, and Junnan Li. Moirai 2.0: When less is more for time series forecasting, orXiv preprint arXiv:2511.11698, 2025a.
Xu Liu, Juncheng Liu, Gerald Woo, Taha Aksu, Yuxuan Liang, Roger Zimmermann, Chenghao Liu, Silvio Savarese, Caiming Xiong, and Doyen Sahoo. Moirai-moe: Empowering time series foundation models with sparse mixture of experts. arXiv preprint arXiv:24 10. 10469, 2024.
Yong Liu, Guo Qin, Zhiyuan Shi, Zhi Chen, Caiyin Yang, Xiangdong Huang, Jianmin Wang, and Mingsheng
Long. Sundial: A family of highly capable time series foundation models. arXiv preprint arXiv:2502.008) 2025b.
Paolo Mancuso, Veronica. Piccialli, and Antonio M Sudoso. machine learning approach for forccasting hierarchical time series. Expert Systems with Applications, 182:115102, 2021.
Yuqi Nie, Nam H Nguyen, Phanwadee Sinthong, and Jan Kalchbrenner. A time series is worth 64 words: Long-lerm forecasting with transformers. arXiv preprint urXie:2211.14733 2023.
Boris N Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. N-beats: Neural basis expansion
analysis for interpretable time series forecasting. In International Conference on Learning Represenlations, 2019.
Youngsuk Park, Danielle Maddix, François-Xavier Xavier Aubet, Kelvin Kan, Jan Gasthaus, and Yuyang Wang. Learning quantile functions without quantile crossing for distribution-free time series forecasting. In International conference on artificial intelligence and slatistics, pp. 8127 8150. PMLR, 2022.
David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Jamuschowski. Deepar: Probabilistic forecasting with autoregressive recurrent networks. International Journal of Forecasting, 36(3):1181 1181)1111 2020.
Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixtaree-of-experss layer. arXiv preprint arXiv:1701.06338 2017.
Xiaoming Shi, Shiyu Wang, Yuqi Nie, Dianqi Li, Zhou Ye, Qingsong Wen, and Ming Jin. Time-moe: Billion- scale time series foundation models with mixture of experts. arXiv preprint arXic:2,09. (00009166444 16040, 2024.
Sean J Taylor and Benjamin Letham. Forecasting at scale. The American Statistician, 72(1):37 45, 2018.
Gerald Woo, Chenghao Liu, Akshat Kumar, Caiming Xiong, Silvio Savarese, and Doyen Sahoo. Unified training of universal time series forecasting transformers, 2024. URL httpp///aaxxw..or/aa///////s
02592
Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer: Decomposition transformers with auto-correlation for long-tery tert series forecasting. In Advances in Neural Information Processing Systems, volume 34, pp. 2D419222439 2021.
Haixu Wu, Tengge Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long. Timesnet: Temporal 2d-variation modeling for general time series analysis. In International Conference Learning Represen- tations, 2023.
Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? arXiv preprint arXic:2205.1 1300000..155,,. 13504, 2023.
Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang. In- former: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 11106-11115, 2021.
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Published in Transactions on Machine Learning Research (03/2026)
Yanqi Zhou, Tao Lei, Hanxiao Liu, Nan Du, Yanping Huang, Vincent Zhao, Andrew M Dai, Quoe V Le, James Landon, et al. Mixture-of-xxeert with expert choice routing. Advances in Neural Information Processing Systems, 35:7103 7114, 2022.
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A Experimentation Setup
We conducted all experiments using 4X NVIDIA A100 80GB GPUs. These resources were utilized primarily for deployment and execution of the TSFMs Moirai small waaall/mdiummmaan /lagge, Sundial, Toto, and Moirai-2.
Initialization window Wint is established by one backtest window on the input, with the window size matching that of forecast horizon T, which subsequently used for weight initialization. For the predictive sampling stage of SYNAPSE arbitration, according to their weights for the mixture of constituent predictive distribution.
total of Monal 1500 for the final mixture, and models contribute
B Evaluation Metrics
To compare the performance of SYNAPSE to that of different. baselines, we primarily use two metries: Mean Absouute Scaled Error (MASE) and Continuous Ranked Probability Score (CRPS). These metrics are defined as below:
Continuous Ranked Probability Score (CRPS): To evaluate the entire predictive distribution. CRPS measures the integrated squared difference between the forecast's cumulative distribution function (CDF) and the empirical CDF the ground truth outcome. Exact computation of CRPS however can be computationally expensive. Therefore, weighted quantile loss (Park et al. 2022 discrete sum over finite set of quantile levels can be used approximation.
CRPPSpprox
where K is the number of quantile levels (e.g., K = 9 for levels {0.1,0.2,.... .., 0.9)). The weighted quantile loss for a single level a is a normalized version of the pinball loss:
wQL(a] 2. P. Mer
Here, Ye is the ground truth value, (a) is the predicted value for the a-th quantile, and Pa is the standard pinball loss function, defined as:
(a( = [oly - ④) yf
1 (1-a) 9) if y
A lower CRPS indicates more accurate and well-calibrated probabilistic forecast.
Mean Absolute Scaled Error (MASE): To provide scale-independent measure of point forecast accuracy, the Mean Absolute Scaled Error (MASE). MASE normalizes the forecast's Mcan Absolute Error (MAE) by the in-sample MAE of naive, one-step seasonal benchmark. This allows for meaningful comparison of accuracy across diverse time series, regardless of their original scale. Formally, MASE is defined as:
MAE MAEforecast
MASE
MAE maive, scasonal
C Domain-Level Performance Comparison of TSFMs vs Oracle
To further quantify the potential gains from dynamic arbitration, we present a granular performance break- down each constituent TSFM across the distinct domains of the GIFT-Eval. As shown in Table even the single best-performing model for specific domain consistently and substantially underperforms the Oracle Arbitrator. This performance gap highlights the critical importance of dynamic, timestep-level arbitration over static domain-level model selection.
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Table 4: Domain-wiest wise performance comparison (CRPS / MASE) individual TSFMs against the Oracle Arbitrator. For each domain and metric, the best (lowest) performing individual TSFM is bolded, and the second best is underlined. The Oracle's performance represents the theoretical lower bound achievable by perfectly selecting the best model at each timestep. Note that the Oracle consistently outperforms even the domain-specilic best models by wide margin.
Domain
Metric
Moirai-L
Moirai-B
Moirai-S
Sundial
Toto
Moirai-2
Oracle
CRPS
0.778
0.835
0.834
0.892
0.848
0.745
0.696
Econ/Fin
MASE
0.845
0.905
0.985
0.923
0.828
0.779
0.761
CRPS
0.732
0.725
0.763
0.645
0.628
0.622
0.383
Energy
MASE
1.026
0.998
1.069
0.839
0.876
0.837
0.519
CRPS
0.565
0.549
0.744
0.772
0.467
0.442
0.371
Healthcare
MASE
0.699
0.683
0.848
0.798
0.625
0.600
0.409
CRPS
0.382
0.388
0.413
0.361
0.348
0.368
0.285
Nature
MASE
0.750
0.771
0.807
0.703
0.736
0.755
0.562
CRPS
0.445
0.424
0.442
0.475
0.424
0.414
0.361
Sales
MASE
0.711
0.695
0.731
0.730
0.705
0.689
0.601
CRPS
0.451
0.478
0.551
0.504
0.477
0.479
0.360
Transport
MASE
0.601
0.637
0.731
0.634
0.632
0.620
0.477
CRPS
0.748
0.781
0.777
0.552
0.500
0.510
0.301
Web/CloudOps
MASE
1.125
1.257
1.136
0.695
0.694
0.665
0.477
Pairwise Comparison
To evaluate the performance consistency of SYNAPSE, we conduct a head-to-head comparison against in- dividual constituents and ensemble basclines across all 97 configurations in GIFT-Eval shown in Figure
B Overall SYNAPSE maintains substantially higher win percentage over other baselines. The strong over- all performance numbers paired with strong win percentages effectively demonstrates the robustness and consistency of SYNAPSE in different time series prediction situations.
Medic: MASE
Metric: CRPS
Mokal, onal
PL
6c.99
Syrapes
Syrapse
Sarapon
Syraper
Ounrida Median Eisembble
Sunckl
74.24
15.0%
93.74
9.74
Syrapon Tra
66.0%
35.3%
71.14
28.9%
Syrapes
Monel
71.24
28.5%
100
Peserage 99)
Posorrage (9)
Figure Pairwise Win/Loss performance comparison of SYNAPSE against constituent individual models and Quantile Median/Ment ensemble baselines. Besides strong overall performance, comparative analysis demonstrates the overall consistency of SYNAPSE.
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E
Performance Scaling with Models Detailed Comparison
As explained in Section 5.8 chronologically order the constituent models Moirai-small, Moirai-base. Moirai-large, Sundial, Toto, Moirai-2 and construct model pool of increasing size for arbitration. Table 5 shows the detailed result of model performance scaling comparison with number models vs best performing individual model.
Table 5: Performance comparison SYNAPSE against the Best Single Model baseline. Scores are shown for different model pools. Here Moirai-S,B,L indicales a pool consisting of Moirai-small, Moirai-mediumm, and Moirai-large models.
CRPS
MASE
Models
SYNAPSE
Best Individual Model
SYNAPSE
Best Individual Model
Moirai-S,B
0.602
0.610
0.882
0.901
Moirai-S,B,L
0.577
0.599
0.850
0.875
Moirai-S,B,L. Sundial
0.534
0.559
0.778
0.750
Moirai-S,B,j Sundial, Toto
0.510
0.517
0.741
0.750
Moirai-S,BB B,LL, Sundial, Toto, Moirai-2
0.496
0.516
0.719
0.728
F
Inference Time Complexity Analysis
SYNAPSE works as post-hoc arbitration mechanism, that executes all N constituent Time Series Foundation Models (TSFMss) exactly once in parallel to generate forecasts for the entire horizon T. Subsequently, the "forward simulation" recalculates scalar weights {w,,r} by sampling from pre-computed distributions; no arbitrated output is fed back into a TSFM for re-inference. Consequently, the total time complexity is the sum of the parallel inference latency of the constituent models and the sequential arbitration logic. The time complexity formally decomposod as:
Ctotal = O ( (m0ammmanneerteamaa (max((nferenco(MM)) + O() 7)
F=1
where 7) represents the constant overhead of the arbitration logic per step. Transformer based TSFM inference typically involves quadratic attention complexity O(r²), and the arbitration overhead scales linearly with the forecast horizon (TT).
Empirically, we measure the total wall-clock time for SYNAPSE against the slowest constituent model (the parallel execution floor) across different datasets and horizons. As detailed in Table the results confirm that the base inference cost. the forward simulation introduces an additive, linear overhead.
floor (with complex models like Toto standard datasets), while
Table 6: Wall-clock inference time comparison (in seconds). The "Max Base Lateney" represents the time (aken by the slowest. individual TSFM in the pool (parallel execution floor). The "SyNAPSE Overhead" isolates the additional time required for the dynamic arbitration logic.
Dataset
Horizon (T)
Max Base Latency (s)
Synapse Total (s)
Synapse Overhead (s)
48
25
++1
ETTI/H
10-15
~88
+20
720
55
+37
KDD Cup 2018/D
30
[1
23
F12
Car Parts
12
22
44
+22
M4 Yearly
120
179
+59
In short-horizon scenarios (c.g., ETT1 Short), the arbitration overhead is relatively small in comparison to base latency. As the forecast horizon T increases (see ETT1 Medium/Long), SYNAPSE overhead grows
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linearly in comparison to the base latency. Note that M4 Yearly is substantially larger than other datasets here. Here, this additive latency is the necessary computational investment to enable dynamic arbitra- tion. SYNAPSE utilizes this linear time cost to generate the simulated ground truth that drives its superior prodictive performance while remains practical even for extended horizons.
G
Synapse: Forecast Stability
SYNAPSE arbitrates on constituent TSFM forecasts solely by sampling from their predictive distributions post-hoc and dynamically adjusting weights based on scoring rules. Consequently, the framework exhibits high forecast stability. To quantify this, exccuted SYNAPSE five times subset of the GIFT-Eval benchmark. As reported in Table 7 the standard deviation of the performance metrics across these runs is quite low, showing that the arbitration mechanism results in overall stable performance.
Table 7: Forecast stability analysis of SYNAPSE over 5 independent runs. The extremely low standard deviations for both MASE and CRPS demonstrate that the arbitration mechanism is robust to sampling noise.
Dataset (Domain)
Horizon MASE (Mean + Std. Dev)
CRPS (Mean + Std. Dev)
Car Parts (Sales)
Short
0.8361 0.0001
0.9543 + 0.0002
ETTI/ H/H (Energy)
Long
1.3431 1 0.0001
0.2608 1 0.00005
ETTI/H (Energy)
Medium
1,2440 + 0.0002
0,2497 : 0.00003
ETTI/R (Energy)
Short
0.8193 1 0.0004
0.1778 1 0.0001
KDD Cup 2018/D (Nature)
Short
1.1753 + 0.0002
0.3732 + 0.0001
H Comparison with Mixture of Experts (MoE) Model
While SYNAPSE conceptually related to "dynamic specialization" with Mixture of Experts architectures like Moirai-MoE) (Liu et al. 2024), the two approaches address fundamentally different constraints. MoE models are typically trained end-to-end while SYNAPSE is post-hoc arbitration framework designed to arbitrate over pre-existing, frozen foundational models. This distinction allows SYNAPSE to be model-agnostic;; it can arbitrate between any set of forecasters, including MoE models themselves, thereby leveraging the strongest available tools without the prohibitive computational cost of retraining specialized experts from scratch.
Moirai-MoF, results are not reported to GIFT-Eval leaderboard, making more difficult compare the performance against SYNAPSE. Furthermore, Moirai-MoE trained on LOTSA (Woo et al.) 2024) creating potential data leakage issue for several datasets in GIFT-Eval, making strictly fair zero-shot comparison difficult. Nevertheless, their reported unleaked zero-shot results the BizlTObs (Web/Cloud) dataset offers comparative perspective. Here, SYNAPSE demonstrates substantially stronger performance compared to Moirai-MoE. Specifically, SYNAPSE achieves MASE of 0.264 and CRPS of 0.072, decisively outper- forming both Moirai-MoE Small (MASE 0.298, CRPS 0.081) and Moirai-Mod Base (MASE 0,290, CRPS 0.079). These results suggest that post-hoc arbitration can effectively rival or surpass the performance of end-to-end specialized architectures in zero-shot scenarios, while maintaining the flexibility incorporate such models constituents in the future.
19

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Published in Transactions on Machine Learning Research (03/2026)
Performance Scaling of Synapse with Architecturally Diverse TSFMs
Table 8: Performance scaling with most recent SOTA TSFMs. SYNAPSE arbitrates over diverse pool including FlowState (SSM), Chronos Sencoder-basedl. and TimesFM 2.5 (Decoder-based), consistently outperforming the single best model.
Model / Method
MASE
CRPS
Toto
0.750
0.517
Moirai
0.728
0.516
FlowState
0.726
0.502
Mime5
0.705
0.490
Chronos-2
0.698
0.485
SYNAPSE (w/o Chronos 2)
0.699
0.479
SYNAPSE (All 5 Models)
0.693
0.474
To verify that SYNAPSE scales effectively with model capability, we extend our evaluation to include a pool of most recently proposed SOTA TSFMs. Each model in this enhanced pool are architecturally more distinct: Chronos-2 2 Ansari et al. 2025) is an encoder-only transformer model that frames forecasting as a language translation task via tokenization; TimesFM-2.5 employs patched decoder-only architecture to efficiently process massive contexts; FlowState (Graf et al. 2025) utilizes State Space Models to generate predictions;
Toto (Cohen et al.] 2024 integrates Student-T mixture head to robustly model heavy-tailed distributions; and Moirai-2 (Liu et al. 2025a) implements a multi-token autoregressive decoder for inference speed. As
detailed in Table S SYNAPSE successfully arbitrates among these different architectures to produce a result superior to any single constituent model. Specifically, the full 5-model configuration achieves a MASE of 0.693 and CRPS 0.474, surpassing the strongest individual model in the pool (Chronos-2) by leveraging the complementary strengths. Notably, SYNAPSE without Chronos-2 achieves comparable MASE and better CRPS than Chronos-2 alone. This confirms that irrespective of how base models improve, SYNAPSE adds value beyond those updates, ensuring the framework continues to set new performance standards.
J Qualitative Plots
To offer qualitative perspective of SYNAPSE and its performance trends in challenging forecast scenarios, we plot SYNAPSE forecasts to that of constituent basclines. Figure demonstrates that SYNAPSE follows the ground truth more closely than the constituents. In simpler scenarios where most models perform competitive performance, SYNAPSE improvements are more modest, whereas challenging examples where models tend to disagree more, performance improvements are more substantial (e.g. M4_ "4__Weekyy/shrr We also showcase showcase some less common cases in Figure (h) and (i) from Nature domain, where SYNAPSE MASE slightly higher than some of the constituent models. Upon closer investigation, can that, when none of the models in the pool give meaningful forecast, SYNAPSE cannot offer any improvement and may often lag behind slightly.
20

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Published in Transactions on Machine Learning Research (03/2026)
Figure 7: Qualitative forecasting results comparing SYNAPSE with baselines on the rest-performing series from 7 different domains.
(MASE: 0.549)
(MASE: 0.565)
Ground Truth (Target)
Synapse (MASE: 0.505)
Sundial (MASE: 0.523)
480
Volue
450
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430
Oct 2020
NoV
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aa) Nature: Jena
Weather/D DD/ssort ID 40
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Moiraí2 (MASE: 0.789)
(MASE: 1.023)
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Synapse (MASE: 0.764)
(MASE: 0.870)
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ww
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(b) Web/CloudOps: BizlTObs Application//0//meeenii ID 2
Context
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Moirai2 (MASE: 0.431)
Toto (MASE: 0.454)
Ground Truth (Target)
Synapse (MASE: 0.356)
Sundial (MASE: 0.435)
Value
20
15
10
Mar 2018
Apr
May
Time
(c) Sales: Hierarchical Sales// ID 88
21

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Published in Transactions on Machine Learning Research (03/2026)
(Continued) Qualitative forecasting results.
Context
Forecast Start
Moiraiz (MASE: 0.549)
Toto (MASE: 0.505)
Ground Truth (Target)
Synapse (MASE: 0.471)
Sundial (MASE: 0.563)
Valuee
May 2010
Time
(d) Energy: ETT2/15T//ong ID 13
Forecast Start
(MASE: 0.395)
(MASE: 0.390)
Ground Truth (Target)
Synapse (MASE: 0.354)
Sundial (MASE: 0,388)
700
600
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300
5ep
Oct
Nov
2019
Time
(e)
Transport: M DENSE/D/shortt ID 51
Context
Forecast Start
MoiraiZ (MASE: 10.632)
Toto (MASE: 9.346)
Ground Truth (Target)
Synapse (MASE: 8.432)
Sundial (MASE: 12.322)
24000
23000
Jalue
21000
20000
O0f
Dec
(701
Time
(f) Econ/Fin: Fin: M4 Weekly /short ID 248
22

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Published in Transactions on Machine Learning Research (03/2026)
(Continued) Qualitative forecasting results.
Context
Forecast Start
Moiral2 (MASE: 0.699)
Toto (MASE: 0.679)
Ground Truth (Target)
Synapse (MASE: 0.689)
Sundial (MASE: 0.848)
350000
340200
330000
Value
320000
300000
290000
Inn
1987
1990
Time
(k) Health US Births BeachssMM/shheeee M/shhrtt ID 1
0.370
Groudd
Harai IAAG HAAGE (.000)
(h) Nature: Jena, Weather/D/short ID 11
sanda (1r0a (ANC.)
Fooocal Cart
2006
(i) Nature: Jena Weather Saather/D/shie D/short ID 31
23


Reviews:
[{"review_id": "9ryMt2yfkk", "reviewer": "Reviewer_3NJQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper evaluates time-series foundation models (TSFMs) in a post-hoc combination scheme. The authors motivate this by stating—and showing (at least under their chosen set of baselines)—that no single TSFM dominates across datasets, domains, and forecast horizons. They also propose an oracle selector that automatically chooses the best model, serving as an upper bound.\n\nTheir main method, Synapse, is an adaptive ensemble/mixture procedure. It constructs an aggregated predictive distribution via weighted predictive sampling from each TSFM’s quantile outputs, using an estimated inverse CDF derived from the quantiles.\n\nThe evaluation is conducted on the popular foundation-model benchmark GIFT-Eval, and the paper compares against a small number of foundation models, mostly based on similar architectures. Stronger current baselines, such as FlowState or Chronos 2, are omitted.\n\n----\n\n**Strengths**:\n\nThe paper addresses an important question: how can we leverage multiple foundation models that are individually insufficient, without requiring training a new model?\n\nPredictive sampling yields a distribution that is probabilistically valid by construction (quantiles computed from an empirical sample are monotone), avoiding ad hoc quantile-crossing fixes.\n\nWithin its limited scope, the paper provides a solid analysis and ablation study.\n\n\n**Weaknesses**:\n\nThe claim that no single model excels is plausible, but a more thorough analysis of the best-performing methods on GIFT-Eval would strengthen it.\n\nClaims of state-of-the-art performance are not convincingly supported given the limited baseline set and the absence of stronger ensemble/combination baselines.\n\nSeveral important implementation details (e.g., window size, initialization, and sensitivity analyses) are underspecified, weakening reproducibility, and there is no released reproduction code.", "claims_and_evidence_yn": "No", "claims_explanation": "I believe the main empirical takeaway—namely, that adaptive combination can outperform single TSFMs and simple ensembles on average, especially at longer horizons—is supported (primarily by Table 1 and Figure 5). However, several claims and key parts of the method description are not yet supported with sufficiently clear, convincing, and reproducible evidence.\n\nThe Figure 1 switching-rate table reports only the percentage of switches, which does not reveal whether one model dominates with occasional flips versus genuinely diverse selection. The radar plot suggests selection frequencies, but it lacks clear axis labeling, which makes magnitude comparisons difficult.\n\nFurthermore, the formulation describes operating on a dynamically selected subset, and the introduction claims subset selection, but Section 3 states that the implementation \"utilizes the full set of models at each step.\" This affects both interpretation (selection vs. weighting) and compute.\n\nThe compute/practicality trade-off is also unclear. The authors state that the experiments were run using 4× A100 80GB GPUs. If Synapse requires running all TSFMs at each horizon step (and potentially feeding arbitrated outputs back autoregressively for some models), deployment cost could be high, yet the paper does not report wall-clock overhead or cost–accuracy trade-offs. The scaling plot only shows that performance improves, but not at what cost. Relatedly, it is unclear what happens when a noisy or suboptimal model is included; such a model could plausibly dampen performance and/or slow inference.\n\nThe introduction states new SOTA and consistently amplifying gains. While long-horizon improvements are visible, SOTA is not established without stronger combination baselines. Moreover, stronger models such as Chronos 2 or FlowState should be included in the experiments.\n\nIn the ablation (Table 3), Sales performs best for the median ensemble and is slightly worse for Synapse variants. This appears to contradict the narrative and warrants further discussion.", "audience_yn": "Yes", "audience_explanation": "The work is relevant and significant to TMLR’s readership. It targets a problem that is increasingly important for the community: practitioners now have multiple strong TSFMs available, yet performance is heterogeneous across domains and forecast horizons. The oracle analysis and benchmark results provide useful evidence that post-hoc combination can help, and Synapse offers a concrete design pattern that many readers will find practical.\n\nHowever, the paper would benefit from a clearer algorithmic specification, stronger baselines, and a more rigorous analysis of the forward-simulation mechanism.", "requested_changes": "**Critical**\n\n* Make the \"selection\" vs. \"weighting over all models\" framing consistent; if the method uses all models each step, state this clearly, and ideally add a \"top-k/budgeted\" variant to control cost.\n* Improve the Figure 1 / oracle diagnostics beyond \"switching rate\" by adding \"top-1 share,\" \"entropy/concentration,\" and \"run-length\" statistics (and/or clearer axis labeling) to distinguish \"dominant with occasional flips\" from \"truly diverse switching.\"\n* Strengthen baselines before using \"SOTA\" language by including at least one stronger combination baseline and a simple \"time-varying loss-weight\" baseline.\n* Report \"wall-clock latency\" and \"cost–accuracy trade-offs,\" and clarify whether inference is \"step-by-step per horizon\" and how compute scales with \"#TSFMs\" and \"horizon.\"\n* The algorithm softmax fallback appears directionally inconsistent with the definition si= AverageCRPS (this we want to minizmize). As written, softmax(s) would upweight higher-loss models. Please clarify what the implementation is using and update the algorithm accordingly.\n\n**Strengthening**\n\n* Discuss when \"forward-simulation weighting\" helps vs. hurts, ideally with a small diagnostic or sensitivity analysis.\n* Explain the \"Sales\" ablation outcome\n* If you are positioning this as multi-specialist cooperation, consider citing Many Minds, One Goal: Time Series Forecasting via Sub-task Specialization and Inter-agent Cooperation (Huang et al., NeurIPS 2025).\n\n**Minor**\n\n* Reduce repetition: \"Experiments 4\" and \"Experimental Setup: Baselines\" largely repeat the same baseline enumeration.", "broader_impact_concerns": "There are no concerns", "additional_comments": "Naming: \"arbitration framework\" is not standard ML forecasting terminology for what is essentially a dynamic ensemble/mixture; consider renaming it or justifying the term with precedent.\n\nMissing space:\nseries foundation models(Das et al., 2024; Graf et al., 2025; Woo et al., 2024; Cohen et al., 2025; Shi et al., 2024).\n\nToto (Cohen et al., 2024), Sundial(Liu et al., 2025b), Moirai2, Moirai-small, Moirai-base, and Moirai-large variants (Woo et al., 2024). (Page 7)", "extra_scores": {}, "extra_text": {}}, {"review_id": "YLMlVrh5Wf", "reviewer": "Reviewer_jvut", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- The authors introduce Synapse, a tool for dynamically utilizing a pool of TSFMs, assigning and adjusting prediction weights based on their relative, context-dependent performance, and creating a robust prediction distribution by adaptively sampling from the output quantiles of the individual models.\n- I like figure 1, and it serves as motivation (how often to change the model for the best possible prediction). The paper could be strengthened by adding how great the performance loss would be without the many changes.\n- The idea presented sounds logical, interesting, and shows good results.\n- I like Figure 2. It reinforces the authors' motivation, but I also find the comparison somewhat unfair at this point, as one model is being compared with the best possible prediction out of different models (oracle). I think that an ensemble would be more appropriate or should be added.\n- I find the paper generally easy to understand. The dataset and models seem appropriate. Sections 4.5 and 4.6 substantiate the paper.", "claims_and_evidence_yn": "Yes", "claims_explanation": "In general, the claims were substantiated by reasoning and experiments, but I have two points that I find insufficient:\n\n- \"This persistent and high-frequency switching underscores that no single model is optimal for an entire forecast window and provides a strong empirical justification for our dynamic arbitration approach, as a static model selection would be consistently suboptimal.\" I understand the motivation here, and in particular that one would like to have a model that predicts as accurately as possible across all data. However, I am missing a comparison where each model was evaluated individually on each domain and horizon, without merging results, in order to justify the statement. It would be possible to determine which model performs best on small subsets of the data and then use that model.\n\n- \"This demonstrates that the synergistic combination of expertise within Synapse is more powerful than relying on any single, top-performing TSFM.\" & \"Critically, we demonstrated that an arbitrated mix of weaker models, when guided by Synapse, can decisively outperform a stronger, single monolithic TSFM\" I don't think this statement was shown here, because Figure 6 does not compare against a stronger model, but against the strongest in the pool currently being considered (i.e., a model that is also part of Synapse). According to Figure 6, Toto and Moirai-2 alone would be stronger than Synapse using the four weaker models.", "audience_yn": "Yes", "audience_explanation": "Yes, I think it's an interesting and timely topic.", "requested_changes": "Critical changes:\n- I lack a comparison of how computationally intensive Synapse is compared to baselines, especially in terms of inference time.\n- I understand the difference to MoE methods, as these are learned end-to-end, but the motivation is similar and, especially during inference, they are more efficient. I would also add an approach as a baseline.\n- In some cases, it makes sense to put related work at the end, but I wouldn't do that here, as it would help the reader understand at the beginning and directly clarifies why these models were chosen.\n- The qualitative plots in Appendix D should be described.\n- It was mentioned in the appendix that N was set to 1500. I am missing an ablation at this point, as this value seems high to me, and a comparison of the inference times would also be interesting.\n\n\nMinor changes:\n- I would add Oracle to Table 1 to see how big the performance gap is.\n- Figures 4 and 5 merely present the results from Table 1 in a different way. I would make better use of the space in the main paper and move these plots to the appendix.\n- References to the appendix would be useful, not just to B.\n- The labelling of the plots should be enlarged.", "broader_impact_concerns": "NA", "additional_comments": "- Is there no research on dynamic arbitration? Is the idea of going beyond uniform distribution, as in the ensemble, unique to this paper?", "extra_scores": {}, "extra_text": {}}, {"review_id": "mwkL5xgVM4", "reviewer": "Reviewer_K563", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper first identifies the representational compromise inherent in monolithically trained models as a key limitation in non-stationary time-series forecasting. To address this issue, the authors reformulate the forecasting task and introduce a dynamic arbitration problem among time-series forecasting models (TSFMs). Furthermore, they propose SYNAPASE, a novel method capable of dynamically selecting a small subset of models and adaptively arbitrating their contributions at each time step. Extensive experimental results demonstrate that SYNAPASE consistently achieves state-of-the-art performance with substantial improvements over existing methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes the claim is supported by experimental results and corresponding analysis.", "audience_yn": "Yes", "audience_explanation": "It is relevant to time series data analysis community.", "requested_changes": "1. In table 3, the authors are suggested to add the error bars for the major results. Statistic significant results are very important to justify the conclusion.\n\n2. The authors have only compared their methods with simple ensemble methods, e.g., mean ensemble and median ensemble, how about the performances of mixture of experts methods? (e.g., some MOESs also adopte expert choice routine [1])\n\n[1] Zhou Y, Lei T, Liu H, et al. Mixture-of-experts with expert choice routing[J]. Advances in Neural Information Processing Systems, 2022, 35: 7103-7114.\n\n3. Lack of discussion regarding the efficiency of the proposed approach. More details regarding the training and inference overheads should be described in Table 1, e.g., number of parameters, GFLOPS and inference time.\n\n\n4. The authors are encouraged to add more failure case analysis in the paper to enbale more insights discussion regarding the limitation of the proposed approach.\n\n5. TSNE visualizations of the embedding spaces of different models are very interesting to be demonstrated.", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)