DistDF: Time-series Forecasting Needs Joint-distribution Wasserstein Alignment
Authors: Eric Wang
Licheng Pan
Yuan Lu
Zhixuan Chu
Xiaoxi Li
Shuting He
Zi Ciu Chan
Qingsong Wen
Haoxuan Li
Zhouchen Lin
Venue: ICLR 2026 Poster (2026)
Keywords: time-series forecasting
Primary area: learning on time series and dynamical systems

Abstract: Training time-series forecast models requires aligning the conditional distribution of model forecasts with that of the label sequence. The standard direct forecast (DF) approach seeks to minimize the conditional negative log-likelihood  of the label sequence, typically estimated using the mean squared error. However, this estimation proves to be biased in the presence of label autocorrelation.  In this paper, we propose DistDF, which achieves alignment by alternatively minimizing a discrepancy between the conditional forecast and label distributions. Because conditional discrepancies are difficult to estimate from finite time-series observations, we introduce a newly proposed joint-distribution Wasserstein discrepancy for time-series forecasting, which provably upper bounds the conditional discrepancy of interest. This discrepancy admits tractable, differentiable estimation from empirical samples and integrates seamlessly with gradient-based training. Extensive experiments show that DistDF improves the performance diverse forecast models and achieves the state-of-the-art forecasting performance. Code is available at https://anonymous.4open.science/r/DistDF-F66B.

Published as a conference paper at ICLR 2026
DISTDF: TIME-SERIES FORECASTING NEEDS WASSERSTEIN ALIGNMENT
Hao Wangh2 Licheng Pan'
Yuan Lu
Zhixuan Chu
Xiaoxi Li'
Shuting He*
Zhichao Chen Qingsong Wen"
Haoxuan Li
Zhouchen Lin
Xiaohongshu Inc. Cooleee of Control Science and Technology. Zhejiang University
s Coollee of Computer Science and Technology. Zhejiang University
4 "School of Computing and Artificial Intelligence, Shanghai University Finance and Economics State Key Lab of General AI, School of Intelligence Science and Technology, Peking University "Squireel AI "Center for Data Science, Peking University 8 IInttitute for Artificial Intelligence, Peking University
ABSTRACT
Training time-series forecast models requires aligning the conditional distribution of model forecasts with that of the label sequence. The standard direct forecast (DF) approach resorts to minimizing the conditional negative log-likelihood of the label sequence, typically estimated using the mean squared error. However, this estimation proves to be biased in the presence of label autocorrelation. In this paper, we propose DistDE, which achieves alignment by alternatively min- imizing a discrepancy between the conditional forecast and label distributions. Because conditional discrepancies are difficult to estimate from finite time-series observations, introduce newly proposed joint- dintriisttibttion Wasserstein dis- crepancy for time-series forecasting, which provably upper bounds the conditional discrepancy of interest. This discrepancy admits tractable, differentiable estimation from empirical samples and integrates seamlessly with gradient- based training. Extensive experiments show that DistDF improves the performance of diverse forecast models and achieves the state-of-the-ar forecasting performance. Code is available https: /ttgt/tttbb//tt/ com Ma Master--a- PLC/DistDS
INTRODUCTION
Time-series forecasting, which entails predicting future values based on historical observations, plays a critical role in numerous applications, such as stock trend analysis in finance (Li al., 2025a), website traffic prediction in e-commerce (Chen al., 2023), and trajectory forecasting in robotics (Fan et al., 2023). In the era of deep learning, the development of effective forecast models hinges two aspects (Wang et al., 2025f): (I) How to design neural architecture serving as the forecast models? and (2) How to design learning objective driving model training? Both aspects are essential for achieving high forecast performance.
The design of neural architectures has been extensively investigated in recent studies. A central challenge involves effectively capturing the autocorrelation structures inherent in the input sequences. To this end, a variety of neural architectures have been proposed (Wang et al., 2023b; Lin et al., 2024). Recent discourse emphasizes the comparison between Transformer-basee self-attention mechanisms to capture autocorrelation and scale effectively (Nie et al., 2023; Liu et al., 2024; Piao et al., 2024) and linear models, which use linear projections model autocorrelation and often achieve competitive performance with reduced complexity (Yi et al., 2023b; Zeng et al., 2023; Yue et al., 2025). These developments illustrate a rapidly evolving aspect time-series forecasting.
models--hichh leverage
In contrast, the design of learning objectives remains comparatively under-explored (Li et al., 2025c; Qiu et al., 2025a; Kudrat et al., 2025). Current approaches typically define the learning objective by estimating the conditional likelihood of the label sequence. In practice, this is often implemented
1 Correspnniing authors.

---

Published as a conference paper at ICLR 2026
as the mean squared error (MSE), which has become standard objective for training forecast models (Lin et al., 2025). However, MSE neglects the autocorrelation structure of the label sequence, leading biased likelihood estimation (Wang et al., 2025g). Some efforts transform the label sequence into conditionally decorrelated components to eliminate the bias (Wang et al., 2025f:g). Nevertheless, as demonstrated in this work, such conditional decorrelation cannot be guaranteed in practice; thus, the bias persists. Therefore, likelihood-baa based methods are fundamentally limited by biased likelihood estimation that impedes model training.
To bypass the limitation of previous widely used likelihood-based methods, we propose Distribution- aware Direct Forecast (DistDF), which trains forecast models by minimizing the discrepancy between the conditional distributions of forecast and label sequences. Since directly estimating conditional discrepancies is intractable given finite time-series observations, we introduce the coint-distribution Wasserstein discrepancy for unbiased time-series forecasting. It upper-bounds the conditional discrep- ancy of interest, enables differentiation, and can be estimated from finite time-series observations, making it well-suited for integration with gradient-based optimization of time-series forecast models.
Our main contributions are summarized as follows:
We demonstrate a fundamental limitation in prevailing likelihood- -baee learning objectives for time-series forecasting: biased likelihood estimation that hampers effective model training.
We propose DistDF, training framework that aligns the conditional distributions of forecasts and labels, with newly proposed joint-distribution Wasserstein discrepancy, ensuring the alignment of conditional distributions and admitting tractable estimation from finite time-series observations.
We perform comprehensive empirical evaluations to demonstrate the effectiveness of DistDF, which enhances the performance of state-of-the-art forecast models across diverse datasets.
2
PRELIMINARIES
2.1 PROBLEM DEFINITION
In this paper, we focus the multi-step time-series forecasting problem. We use uppercase letters (e.g., X) to denote matrices and lowercase letters (e.g., x)) to denote scalars. Given a time-series dataset S with D covariates, the historical sequence at time step is defined X (S,-H+1,..... S,, RExDD, and the label sequence is defined as Y [S.+ S.+ E RT where H is the lookback window size and T is the forecast horizon. Modern models adopt a direct forecasting (DF) approach, generating all T forecast steps simultaneously (Liu et al., 2024). Thus, the target is to learn a model H + T that maps X to forecast sequence Y approximating Y
The development of forecast models encompasses two principal aspects: (1) neural network archi- tectures that effectively encode historical sequences (Zeng et al., 2023; Liu et al., 2024), and (2) learning objectives for training neural networks (Wang et al., 2025f;g). is important to emphasize that this work focuses on the design of learning objectives rather than proposing novel architectures. Nevertheless, we provide concise review of both aspects for contextual completeness.
2.2 NEURAL NETWORK ARCHITECTURES IN TIME-SERIES FORECASTING
Architectural developments aim to encode historical sequences to obtain informative representa- tion (Wu et al., 2025; Qiu et al., 2025b). Representative classic architectures include recurrent neural networks (Gu et al., 2021), convolutional neural networks (Luo and Wang, 2024), and graph neural networks (Yi et al., 2023a). A central theme in recent literature is the comparison of Transformer and non- con-Trassformet architectures. Transformers (e.g., PatchTST (Nie et al., 2023). TQNet (Lin et al., 2025), TimeBridge (Liu et al., 2025)) demonstrate strong scalability on large datasets but often entail substantial computational cost. In contrast, non-Traasformer models (e.g., TimeMixer (Wang et al., 2024), FreTS (Yi et al., 2023b)) offer greater computational efficiency but may be less scal- able. Recent advances include hybrid architectures that combine Transformer and Transforsfrr components for their complementary strengths (Lin et al., 2024), as well as the integration of Fourier analysis for efficient learning (Piao al., 2024; Yi al., 2025).
'ereaattrr, consider the univariate case (D) 1) for clarity. In the multivariate case, each variable can be treated a separate univariate case when computing the learning objectives.
2

---

Published as a conference paper at ICLR 2026
2.3 LEARNING OBJECTIVES IN TIME-SERIES FORECASTING
Learning objective developments have largely focused on aligning the conditional distributions of model forecasts P(Y|X) x with those of the label sequence X. To this end, the most common objective is the MSE, which measures the point-wise error between the forecast and label sequences (Dai et al., 2024; Chen et al., 2025; Lin et al., 2025):
Vix - 1 Σ
-(ixx )ix.)) (Mx))
Lusse
where Yjx is the label sequence given historical sequence X, exx is the forecast sequence. However, the MSE objective is known to be biased since it overlooks the presence of label autocorrelation (Wang et al., 2025g). To mitigate this issue, several alternative learning objectives have been proposed. One line of work advocates aligning the overall shape of the forecast and label sequence (e-g., Dilate (Le Guen and Thome, 2019) and PS (Kudrat et al., 2025)). These approaches accommodate au- tocorrelation by emphasizing sequence-level differences, but lack theoretical guarantees for achieving an unbiased objective. Another line of work transforms labels into decorrelated components before aligmment. This strategy reduces bias and improves forecasting performance (Wang et al., 2025f:g), showcasing the benefits of refining learning objectives for time-series forecasting.
METHODOLOGY
3.1 MOTIVATION
The primary objective in training time-series forecast models is to align the conditional distribution of model-generated forecasts with that of the label sequence. Likelihood-basee approaches seek this by maximizing the conditional likelihood of the label sequence. common practice is to estimate the negative liglliiel through the mean squared error (MSE), which has become the predominant objective for training time-series forecast models (Lin et al., 2025). However, MSE treats each future step as an independent prediction task and thus ignores the autocorrelation structure of the label sequence, where each observation typically depends on its predecessors (Zeng et al., 2023). Such an oversight renders MSE biased from the true negative log- likelihood of the label sequence. This issue is termed as autocorrelation bias and formalized in Theorem 3.1.
Theorem 3.1 (Autocorrelation bias). Suppose YIx E RTT is the label sequence given historical sequence X, Yix= IT is the forecast sequence, 21x E RTXX is the conditional covariance of Y xx- The bias of MSE from the negative log-likelihood of the label sequence given X is expressed as:
Bias- = | - 8|x T
- ||xix fixx-ixx1)
81x
where |1012 2 = vT E v. It vanishes if the conditional covariance Σ|x is the identity matrix?.
Some might argue that the bias can be eliminated by first transforming the label sequence into conditionally decorrelated components and then applying MSE component-wise For example, FreDF (Wang et al., 2025g) uses Fourier transform to obtain frequency components; Time-ol (Wang et al., 2025f) employs principal component analysis to obtain principal components. This strategy does eliminate the bias if the resulting components were truly conditionally decorrelated (see Theo- rem 3. However, one key distinction warrants emphasis: both Fourier and principal component transformations guarantee only marginally decorrelated of the obtained components (i.e., diagonal Σ). not the required conditional decorrelation (i.e., diagonal Σjx)': thus the bias persists. Hence, likelihood based methods are limited by biased likelihood estimation which hampers model training.
"The pioneering work (Wang al., 20251) derives the bias from the marginal likelihood of Y assuming it follows Gaussian distribution. In contrast, this work clarifies that is the conditional distribution of Y given X that is Gaussian. Consequently, we derive the hias from the conditional log- likelihood Y. "According to Theorem 3.3 (Wang et al., 2025g) and I Lemma 3.2 (Wang al., 2025f), the components obtained by Fourier and principal component transformations are marginal decorrelated.
3

---

Published as a conference paper at ICLR 2026
1.000
1.000
1,000
32
32
0.100
0.100
0.100
84
0.000
0.010
0.010
128
0.001
0.001
0.001
160
-0.000
192
-0.000
-0.000
81
182
(a) Raw labels.
(b) FreDF components.
(c) Time-ol components.
Figure 1: The conditional correlation of label components given X, where the forecast horizon is set to T = 192. The correlation matrices are computed for the raw labels (a), the frequency components FreDF (b) (Wang et al., 2025g) and the principal components in Time-ol (c) (Wang et al., 2025().
study the Traffic dataset illustrate the limitations of likelihood- based methods. As shown in Fig. ((a), the conditional correlation matrix reveals substantial off- diagonal values-ove 50.3% exceed 0. iillstttta the presence of autocorrelation effects. In contrast, Fig. l(b) presents the conditional correlations of the latent components extracted by FreDF and Time-ol (Wang et al., 2025g:f). While the non-diagonal elements are notably reduced, residual correlations remain, indicating that these methods do not fully eliminate autocorrelation in the transformed components. Consequently, applying a point-wise loss these transformed components continues to ignore autocorrelation and yields bias.
Case study. We conduct
Given the substantial challenges faced by likelihood- methods, it is worthwhile to explore alternative strategies to align conditional distributions for model training. One plain strategy is directly minimizing a distributional discrepancy between the conditional distributions (Courty et al., 2017), which effectively achieve alignment while bypassing the complexity of likelihood estimation. Importantly, there are two questions that warrant investigation. How to devise a discrepancy to align the conditional distributions? Does it effectively improve forecast performance?
3.2 ALIGNING CONDITIONAL DISTRIBUTIONS VIA JOINT- CONT-DISTTIBBTION BALANCING
In this section, we aim to align the conditional distributions, i.e., P y and Py|x, x by minimizing discrepancy metric between them. As with general distribution alignment tasks, the choice of discrepancy metric crucial (Xu et al., 2021). We select the Wasserstein discrepancy from optimal transport theory, which measures the discrepancy between two distributions as the minimum cost required to transform one into the other. Its ability to remain informative for distributions with disjoint supports, combined with its robust theoretical properties and proven empirical success, makes a principled choice for this work (Courty et al., 2017). An informal definition is provided in Definition 3.2.
Definition 3.2 (Wasserstein discrepancy). Let a and B be random variables with probability distri- butions Pa and 33 S., [ai,..., anl and S3 [3,, Bm)] be empirical samples from Pa and F The optimization problem seeks a feasible plan P E R to transport a to 3 at the minimum cost:
WPPPP, = min PE
(D,P),
Pi + ... P Pr.m a,, 1, I,......
(3)
IIP..,, Pi.j + + Pn.j bj.j 1,... M,
Paj 2 0, I,... n, = 1,....
where Wp denotes the p- Wasserstein discrepancy; D E Ruxm represents the pairwise distances calculated as Dij = ||a, - 3311"; a = (ai...., an) and = bb...., bm) are the weights of samples in a and B, respectively; n and m are the numbers of samples; 11 defines the set of constraints.
natural approach to aligning the conditional distributions is to minimize the Wasserstein discrepancy (Py II x However, this approach suffers from estimation difficulty. For any given X, typical dataset often provides only single associated label sequence Y, and the forecast model
4

---

Published as a conference paper at ICLR 2026
produces only single output Y. Thus, the empirical sets (Syjx and Spix) each contain only single sample, which is insufficient represent the underlying conditional distributions and renders the discrepancy uninformative. Crucially, this limitation is not unique to the Wasserstein discrepancy; any distributional discrepancy metric becomes degenerate in the absence of multiple samples.
Lemma 3.3 (Kim et al. (2022)). For any p Z L, the coint-iittribution Wasserstein discrepancy upper Wasserstein discrepancy:
bounds the expected
f P
"x,(Px.y,Pxxx)))
(4)
where the equality holds p = 1 or the conditional Wasserstein term is constant with respect to X.
To bypass this estimation difficulty, we advocate the joint-distribution Wasserstein discrepancy, (P,,,, P x,y,, for training time-series forecast models. This proxy is advantageous for two reasons. First, it provides a provable upper bound on the expected conditional discrepancy (see Lemma 3.3), ensuring that minimizing the joint discrepancy effectively aligns the conditional dis- tributions of interest. Second, it is readily estimable from finite time-series observations, since the empirical samples Sx,y and Sx,y can be constructed from the entire dataset, yielding sufficient samples to compute meaningful and informative discrepancy.
Theorem 3.4 (Alignment property). The conditional distributions are aligned, i.e., Py|x Pyxxx if the point-distributionn Wasserstein discrepancy is minimized to zero, i.e., (P
=
Lemma 3.5 (Peyré and Cuturi (2019)). Suppose Pxy and P xx obey Gaussian distributions (px,, Ex,y) and (uxxy. Ex,y), respectively. The squared W2 discrepancy can be calculated as the Bures- Wassersessrr discrepancy:
DW(xxy,Xx., Exy, Ex,y) |X.X - 1x,8 1, +B(Ex,,,xx,)) -
(5)
where B(xxx,, =x,)==11 IT (xXx + Exx 2 2
1/22 x.y
Tr(-) denotes matrix trace.
Theoretical Justification. Theorem 3.4 shows that minimizing the joint-distribation Wasserstein discrepancy zero guarantees the alignment of conditional distributions. This result enables using the joint discrepancy as learning objective for training forecast models. Under a Gaussian assumption (likewise MSE), this discrepancy has an analytical form (Lemma 3.5), obviating the need to solve the complex transport problem of Definition 3.2. The proof is available in Appendix A.
The use of Wasserstein discrepancy for distribution alignment is highly inspired by domain adaptation field (Courty et al., 2017). However, one key distinction warrants emphasis. Domain adaptation dominantly aligns the marginal distributions of inputs to improve generalization; in contrast, we align the conditional distributions of model outputs and labels to perform supervised training. To our knowledge, this represents a technically innovative strategy.
3.3 MODEL IMPLEMENTATION
In this section, present the implementation specifics of DistDE, a framework that leverages the joint-distribution Wasserstein discrepancy to enhance the training of time-series forecast models. The principal steps of the algorithm are formalized in Algorithm 1.
Given historical sequences X and corresponding label sequences Y E RBxT where B denotes batch size and T denotes forecast horizon; the forecast model g is employed to generate the forecast sequences, denoted as Y (step 1). Subsequently, we define two two joint sequences, which are constructed by concatenating X with Y and Y along the time axis, respectively (step 2), expressed as [X,Y] and 2 [X,
To quantify the discrepancy term Cdist, com- pute the first- and second-order statistics Z and 2. iee. the mean vectors (Hz and 22) and covariance matrices (Ez and Σ,2) (steps 3-4).
Algorithm 1 The workflow of DistDE.
Input: X: historical sequences, Y: label sequences. Parameter: the relative weight of the discrepancy, g: the forecast model to generate forecast sequenee. Dutputt La: the obtained learning objective.
1: + o(X)
2: f concated X. Y). 2 concate X. Y)

---

Published as a conference paper at ICLR 2026
Table Long-term forecasting performance.
DisDF (Ours)
Timellnidge (3025)
l'redformer (2024)
ilrandformer
PreTS
TimesNer
MICN
TIDE
PatchTST
Diimeer
Model
(2024)
(2023)
(2023)
(2033)
(2023)
(2013)
(2023)
Mecris
MSE
MAB
MII
MAB
MSE
MAE
MSE
MAB
MSE
MAE
MSB
MAU
MSC
MAU
MSE
MAL
MSU
MAE
MSE
MAB
FT7ml
0.378
0.394
0.887
0 4000
D. 382
108
0,411
0:14
0-44
0.431
0.438
0.400
00.99
0.431
04415
0:407
0.389
0.400)
0.403
0:407
B77m2
0.277
0,321
0.281
0.326
0.200
0.524
0.205
0.336
0.316
0.365
0,302
0.354
00.00
0.364
0.236
0.325
0.303
0.344
0.342 0.392
ETTbl
0.429
0.442
0.447
0.434
0.452
0.448
0.499
0.472
0.463
0.533
0.448
0:435
0,459
0L451
0.456
61112
0,893
0.402
0,400
0.524
0.420
0.599
FCT.
0.367
0.116
0.191
0.284
[0.179
0.270
(0119)
0.288
0.212
0.306
[0:191
0.215
0.390
0.:95
0.286
(2212
Trafle
8.417
0,279
0:426
0.262
0.466
0.536
0.426
0.285
0.538
0.380
0.631
0.358
[0.529
0.312
0.624
0.373
0.468
0.299
(.625 0.384
Weathr
0.248
0.275
0,252
0.277
0.261
0.282
0.369
0.289
0.269
0.293
0.271
0.295
00264
0.521
0.272
0.291
0.267
0.258
0.265
0.317
PEMSOS
0.104
0,315
0.112
0.146
0.200 0.123
0223
0.149
0.126
0.230
0.106
0.223
0.516
0.370 0,870
0.252 0.216
0.223
0171
02249
0.114
0.152
0.343
0.318
0.201 0.908
Note: We the input length following Liu (2021). Bold and undertined denote best and second best results, respectively. Aug indicates erage over botizons: 192, and DistDF' employs the top performing baseline on each dataset as underfying model.
The discrepancy term Ldist is then evaluated using the Sunes-Wasserstein metric (step 5), defined in Lemma 3.5.
Given the complexity of directly optimizing the Busss-Was discrepancy and its lack of inherent pairing awareness, we integrate it with the mean squared error to promote training sta- bility and facilitate convergence (steps 6-7), fol- lowing the established practices (Wang et al., 2025f:g):
L. == a Ldist + (1 (1- (1-a) a) -mse-
(6)
where 0 S a S 1 balances contribution of the distributional discrepancy term.
By minimizing the distributional discrepancy, DistD effectively aligns the conditional distributions of the forecast and label sequences, thereby refining the model's forecast performance. DistDF preserves the principal benefits of the canonical DF framework (Zeng et al., 2023; Liu et al., 2024), such as efficient inference and multi-task learning capability. Moreover, DistDF is model-agnostic, which renders it plugin-and and-play component to improve the training of different forecast models.
4 EXPERIMENTS
To demonstrate the efficacy of DistDF, the following aspects deserve empirical investigation:
Performance: Does DisiDF perform well? Section 4.2, we benchmark DistDF against state-of-
the-art baselines, and in Section 4.3, we compare it with alternative learning objectives.
Gain: Why does it work? In section 4.4, we perform an ablative study, dissecting the individual components of DistDF and clarifying their contributions to forecast accuracy.
3. Generality: Does it support other models and discrepancy measures? In Section 4.5, we examine its compatibility with various models and discrepancies, with further results in Appendix D.4.
4. Sensitivity: Is it sensitive hyperparameters? In Section 4.6, analyze the sensitivity of DistDF to the hyperparameter a, showing stable performance across a broad parameter range.
Efficiency: What is the computational cost of it? In Appendix D.7, we evaluate the running cost of DistDF across different scenarios.
4.1 SETUP
Datasets. We evaluate our methods using several standard public benchmarks for long-term time- series forecasting, following Wu et al. (2023). Specifically,
the ETT dataset (four subsets),

---

Published as a conference paper at ICLR 2026
1.2
1.2
Geround Trush
GEoand Trush
0.0
0.0
-0.5
-0.5
M00 W
0.0
-1.0
4..5
10.5
GocundTruh Predictin
-1.
GrcundTuch
Predicton
-12
-12
-20
-200
100
300
300
(a) ET snapshot.
(b) ECL snapshot.
Figure 2: The forecast sequence of DF (in blue) and DistDF (in red), with historical length H 96.
Table 2: Comparative results with other objectives for time-series forecasting.
DisDF
Time-ol
FreDF
Koopmun
Soft ATT-
DE
Metrics
MSF
MAE
Timebrdgee
ETTml
0.383
0.397
0.383
0.397
0.386
0.398
0.450
0.438
0.387
0.400
0.395
0.402
0.387
0.400
ETThl
0.434
0.436
0.439
0.438
0.439
0.436
0.459
0.449
0.464
0.452
0.452
0.445
0.442
0.440
BCL
0.172
0.267
0.175
0.268
0.175
0.267
0.182
0.277
0.176
0.271
0.268
0.176
0.271
Weather
0.248
0.275
0.250
0.275
0.254
0.276
0,269
0.293
0.252
0.277
0.260
0.280
0.252
0.277
Fredfomme
ETTm1
0.378
0.394
0.379
0.393
0.384
0.394
0.389
0.400
0.389
0.400
0.397
0.402
0.387
0.398
ETThi
0.430
0.429
0.431
0.429
0.438
0.434
0.452
0.443
0.453
0.442
0.460
0.449
0.447
0.434
ECL 0.173
0.266
0.178
0.270
0.179
0.272
0.190
0.282
0.280
0.206
0.298
0.191
0.284
Weather 0.255
0.277
0.255
0.276
0.256
0.277
0.257
0.279
0.258
0,280
0.261
0.280
0.261
0.282
Note: Bold and underlined denote best and second-hest recults, respectively. The reported rocalts are averaged over forecast borizon:: T-96, 192, 336 and 720.
When metric valoes coincide to three decimal places, Bold indicates the sumencally supenor result based on full precision.
ECL, Traffic, Weather, and PEMS (Liu et al., 2024). All datasets are split chronologically into training. validation, and test sets. Comprehensive dataset statistics are presented in Appendix C.I.
Baselines. We compare DistDF to range of competitive baselines, categorized as: (1) Transformer- (Nie et al., 2023), iTrrasformer (Liu et al., 2024), Fredformer (Piao et al., 2024) and TimeBridge (Liu et al., 2025); (2) Non-Transformer based models- DI inear (Zeng et al., 2023), TiDE (Das al., 2023), MICN (Wang et al., 2023b) and FreTS (Yi et al., 2023b).
based
Implementation. Baseline implementations closely follow the official codebase from Piao al. (2024). To ensure fair comparison, the drop-last trick is disabled for all models, as recommended in Qiu al. (2024). All models trained with the Adam optimizer (Kingma and Ba, 2015). When integrating DistDF into a baseline forecast model, we retain all hyperparameters from the public benchmarks (Liu et al., 2024; Piao et al., 2024), only tuning a and the learning rate. Experiments are run on Intel(R) Xeon(R) Platinum 8383C CPUs with 32 NVIDIA RTX H100 GPUs. Further implementation details are provided in Appendix C.
4.2 OVERALL PERFORMANCE
Table reports the long-term forecasting results. DistDF consistently enhances the performance of base models across all evaluated datasets. For instance, on ETTh1, DistDF reduces the MSE of TimeBridge by 0.016. Similar improvements observed on other benchmarks confirm its robustness and generalizability. We attribute these empirical improvements to DistDF's ability to align conditional distributions, property supported by its theoretical guarantees (Theorem 3.4).
Showcases. To further illustrate the practical benefits, we compare the forecast sequences of DF and DistDF in Fig. 2. While a model trained with the standard DF objective captures the overall trend, it fails accurately track fine- fine-grained variations, such rapid changes between steps 100 and 200. In contrast, DistDF produces forecasts that more precisely reflect these subtle and rapid changes, highlighting its effectiveness in improving real-world forecasting accuracy.
7

---

Published as a conference paper at ICLR 2026
Table 3: Ablation study results.
Model
Align
Align 32
Data
T=96
T=192
T=336
T=720
Avg
MSB
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
ETTml
0.326
0.361
0.365
0.382
0.396
0.404
0.459
0.444
0.387
0.398
DF
x
ETThl
0.377
0.396
0.437
0.425
0.486
0.449
0.488
0.467
0.447
0.434
0.217
Weather
0.168
0.211
0.214
0.254
0.273
0.297
0.353
0.347
0.252
0.277
ETTml
0.318
0.359
0.361
0.382
0.393
0.404
0.453
0.440
0.381
0.396
DisDDPt
X
ETThi
0.375
0.394
0.435
0.426
0.471
0.446
0.457
0.455
0.435
0,430
0.273
Weather
0.168
0.211
0.213
0.253
0.273
0.296
0.349
0.348
0.251
0.277
ETTml
0.328
0.365
0.364
0.385
0.395
0.406
0.457
0.441
0.386
0,399
DistDF
ETThl
0.374
0.396
0.430
0.430
0.476
0.451
0.476
0.472
0.439
0.437
ECT.
0.141
0.239
0.161
0.257
0.179
0.273
0.216
0.307
0.174
0.269
Weather
0.168
0.211
0.214
0.253
0,270
0.296
0.353
0.342
0.251
0,277
0.381
0.448
DisDF
ETTh1
0.373
0.393
0,428
0.425
0.466
0.445
0.453
0.455
0.430
0.429
ECL
0.137
0.235
0.159
0.257
0.178
0.272
0.212
0.302
0.172
0.267
Weather
0.252
0.348
Rold and underlined denote and sccand-best respectively. sumerically supetior result on full peecision.
colncide decinal
Rold indicates
4.3 LEARNING OBJECTIVE COMPARISON
Table 2 presents a comparison between DistDF and several established time-series learning objec- tives: Time-ol (Wang et al., 20251), FreDF (Wang et al., 2025g), Koopman (Lange et al., 202 I), Dilate (Le Guen and Thome, 2019). Soft-DTW (Cuturi and Blondel, 2017), and DPTA (Sakoe and Chiba, 2003). this comparison, all methods are integrated into both TimeBridge and Fredformer using their official implementations for ensuring fairness.
In general, shape alignment objectives (Dilate, Soft-DTW, DPTA) improve marginally over standard DF, consistent with findings by Le Guen and Thome (2019). This suggests that heuristic shape-level alignment does not guarantee alignment of conditional distributions. FreDF and Time-ol reduce the bias in likelihood estimation and improve performance. However, established in Section 3. residual bias remains, preventing unbiased alignment of conditional distributions. DistDF minimizes the discrepancy between conditional distributions, achieving unbiased alignment with theoretical guarantees (see Theorem3.44 3.4), and consequently delivers superior performance.
4.4 ABLATION STUDIES
Table 3 examines the two components the dintrdiitri Wasserstein discrepancy (5): mean alignment and covariance alignment. The main findings are as follows:
DistDFt augments DF by aligning only the means of the joint distributions, omitting the B(-) in (5). This approach outperforms DF, illustrating that mean alignment of joint distributions can improve the alignment of the conditional distributions between label and forecast sequence.
DistDFi improves DF by aligning only the variance of joint distributions, exclusively involving B(-) in (5). This approach also leads improvements over DF in most cases, illustrating that variance alignment of joint distributions improves the alignment of the conditional distributions.
DistDF combines both mean and variance alignment for comprehensive joint distribution matching. It yields the best results, demonstrating a synergistic effect when both components are integrated.
4.5 GENERALIZATION STUDIES
In this section, we assess the generalizability of DistDF by applying it different distribution discrepancy measures and across various forecast models.
Varying diserepancy. We evaluate alternative discrepancy measures to align the joint distribution and report the results in Table 4. Specifically, consider Kullback-Leibler (KL) divergence, maxi- mum mean discrepancy (MMD) with RBF and linear kernels, and earth mover discrepancy (EMD). All discrepancies show improvements the standard DF, indicating the benefit of incorporating
8

---

Published as a conference paper at ICLR 2026
0.24
0.32
0:34
0.22
-2.2%
53%
0.2%
0:30
0.5%
-1.4%
weo
+1.4%
0.32
0%
0.28
9:18
0:28
0:28
0.18
Tim
Fredd
ITrana
Prets
Time
Fratt
Time
Fred
ITrena.
Prrtt
FTrenes
Fruit
(a) ECL with MSE
(b) ECL with MAE
(c) Weather with MSE
(d) Weather with MAE
Figure 3: Improvement of DistDF applied to different forecast models, shown with colored bars for
means over forecast lengths (96, 192, 336, 720) and error bars for 50% confidence intervals.
Table 4: Comparative results with other discrepancies for aligning the joint distributions.
Discrepancy
Ours
EMD
MMD@
MMD@RBF
KL
DF
Metries
MSF
MAF
MSF
MAF
MSF:
MAF
MSF
MAE
MSF
MAF
MSF
MAF
Timetragge
ETTml
0.383
0.398
0.388
0.400
0.385
0.400
0.387
0.399
0.387
0.400
0.387
0.400
ETThl
0.433
0.437
0,441
0.439
0.438
0.437
0.441
0.440
0.437
0.438
0.442
0.440
ECL
0.267
0.172
Weather
0.248
0.275
0
0.276
0.253
0.278
0.250
0.276
0.253
0.277
0.252
0.277
Frolformcr
ETTml
0.379
0,386
0.397
0,380
0.395
0.385
0.397
0,385
0.397
0.387
ETibl
0.429
0.431
0.445
0.435
0.437
0.452
0.444
0.435
0.444
0.435
0.447
0.434
ECL
0.183
0.275
0.187
0.280
0,188
0.280
0.187
0.280
0.187
0.279
0.191
0.284
Weather
0.261
0.282
0.262
0.282
0.262
0.282
0.261
0.282
0.261
0.282
Bold and underlined denote and second-best results, respoctioelly regorted resolts overaged forecaal horions: T-96, 192, and 720. When metric values coincide up three decimal places, Bold indicates the mumerccllly superior result based on full precision.
distribution alignment for training forecast models. The joint-distributinn Wasserstein discrepancy achieves the best overall results in 14 out of 16 cases, underscoring its effectiveness in reliably aligning distributions (Peyré and Cuturi, 2019).
Varying forecast models. We further demonstrate the flexibility of DistDF by integrating it into several representative models: TrieBridge, FredFormer, iTransformer, and FreTTS. As shown in Fig. 3, DistF consistently enhances forecast performance across all tested models. For example, on the ECL dataset, I'Taansfommer and FredFormer augmented with DistDF achieve substantial MSE reductions--u to 2.7% and 4.3%, respectively. These results highlight DistDF's potential as general, plug-and-play enhancement for supporting various forecast models.
4.6 HYPERPARAMETER SENSITIVITY
In this section, we analyze how varying the weight of the distributional discrepancy influences DistDF's performance, as summarized in Table 5 and Table 6. On the one hand, increasing a from 0 generally leads improved performance; for example, FredFormer achieves 0.01 reduction in MSE on ECL., showcasing the utility of incorporating the discrepancy term into the learning objective of training forecast models. On the other hand, the optimal performance is typically observed for a 1. This underscores the complementary role of MSE, which is both easy to optimize and straightforward for ensuring point-wise accuracy between the forecast and label sequences.
CONCLUSION
In this study, we demonstrate that existing shelihood-basee approaches suffer from biased likelihood estimation. Instead, we propose DistDF, which trains forecast models by minimizing the discrepancy between the conditional distributions of forecasts and labels. Recognizing the intractability of directly estimating conditional discrepancies, we propose the use of joint-distribution Wasserstein discrep- ancy, which serves as a tractable upper bound and can be efficiently estimated from observed data. By minimizing this quantity, DistDF provably aligns the conditional distributions with theoretical guarantees for training forecast models. Extensive experiments corroborate that DistDF consistently yields improvements in forecast accuracy.

---

Published as a conference paper at ICLR 2026
Table Varying a results of TimeBridge
Table 6: Varying results of Fredformer.
E1Th2
ECL
Weathe
ET
ECL
Weather
Or
MSE
MAE
MISE
MAE.
MSI
MAT.
MSE
MAT.
MST
MAT
MSE
MAE.
0.271
0.261
0.282
0.001
0.378
0.402
0.172
0.267
0.250
0.276
0.001
0.371
0.397
0.182
0.275
0.257
0.279
0.002
0.377
0.402
0.173
0.267
0,250
0.276
0,002
0.372
0.398
0.181
0.274
0.257
0.279
0.257
0.01
0.376
0.400
0.172
0.267
0.249
0.276
0.01
0.370
0.397
0.183
0.275
0.257
0.279
0.02
0.576
0,400
0.174
0.269
0,249
0.276
0,02
0.369
0.395
0.182
0.275
0.258
0.280
0.05
0.259
0.1
0.575
0.399
0.174
0.269
0,254
0,280
0.1
0.371
0.397
0.196
0.287
0.261
0.283
0.2
0.376
0.399
0.177
0.270
0.258
0.282
0.2
0.372
0.39%
0.209
0.298
0.263
0.285
0.378
0.400
0,185
0.277
0.261
0.285
0.5
0.376
0.399
0.230
0.317
0.266
0.287
0.402
0.197
0,282
0.265
0.286
0.386
0.406
0.239
0.326
0.268
0.290
None: Hold and uodedlied denose the best and second- best r2sules.
Note: Bold and underlined danete the best and second-best resalts.
Limitations, According to Lemma 3.5, DistDF quantifies the divergence between the mean and covariance of the joint distributions, thereby capturing global distributional properties. However, discards elementwise correspondences between forecast and label sequences information critical for forecasting tasks. Therefore, DistDF is most effective when employed as a regularization term alongside the standard MSE loss, where MSE recovers elementwise correspondences and fully unleashes the potential of the proposed DistD.
ACKNOWLEDGMENTS
Z. Lin was supported by the NSF China (No. 62276004), the Beijing Natural Science Foundation (No. I.257007), and the Beijing Major Science and Technology Project (No. Z251 100008425006).
REFERENCES
Jason M. Altschuler, Jonathan Weed, and Philippe Rigollct. Near-linear time approximation algo- rithms for optimal transport via sinkhorn iteration. In Proc. Adv. Neural Inf. Process. Syst., pages 1964-1974, 2017.
Nicolas Bonneel, Michiel van de Panne, Sylvain Paris, and Wolfgang Heidrich. Displacement interpolation using lagrangian mass transport. ACM Trans. Graph., 30(6):158, 2011.
Hui Chen, Viet Luong, Lopamudra Mukherjee, and Vikas Singh. Simpletm: simple baseline for multivariate time series forecasting. In Proc. Int. Conf. Learn. Represent. 2025.
Zhichao Chen, Leilei Ding, Zhixuan Chu, Yucheng Qi, Jianmin Huang, and Hao Wang. Monotonic neural ordinary differential equation: Time-series forecasting for cumulative data. In Proc. ACM Int. Conf. Inf. Knowl. Manag., pages 4523-4529, 2023.
Zhichao Chen, Haoxuan Li. Fangyikang Wang. Haotian Zhang, Hu Xu, Xiaoyu Jiang, Zhihuan Song, and Hao Wang. Rethinking the diffusion models for missing data imputation: A gradient flow perspective. In Proc. Adv. Neural Inf. Process. Syst., 2024.
Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, and François-Xavier Vialard. Scaling algorithms for unbalanced optimal transport problems. Math. Comput., 87(314):2563-2600 2018.
Nicolas Courty, Rémi Flamary, Devis Tuia, and Alain Rakotomamonjy. Optimal transport for domain adaptation. IEEE Trans. Pattern Anal. Mach. Intell., 39(9): 2017.
Marco Cuturi and Mathieu Blondel. Soft-dtw: differentiable loss function for time-series. In Proc. Ini. Conf. Mach. Learn., pages 894-903. PMLR, 2017.
Tao Dai, Beiliang Wu, Peiyuan Liu, Naiqi Li, Jigang Bao, Yong Jiang, and Shu-Tao Xia. Periodicity decoupling framework for long-term series forecasting. In Proc. Int. Conf. Learn Represent., 2024.
Abhimanyu Das, Weihao Kong, Andrew Leach, Rajat Sen, and Rose Yu. Long-term forecasting with tide: Time-series dense encoder. Trans. Mach. Learn. Res., 2023.
10

---

Published as a conference paper at ICLR 2026
Jiajun Fan, Yuzheng Zhuang, Yuecheng Liu, Hao Jianye, Bin Wang, Jiangcheng Zhu, Hao Wang. and Shu-Tao Xia. Learnable behavior control: Breaking atari human world records via sample-efficient behavior selection. In Proc. Int. Conf. Learn. Represent, pages 1-9, 2023.
Albert Gu, Karan Goel, and Christopher Re. Efficiently modeling long sequences with structured state spaces. In Proc. Int. Conf. Learn. Represent,, 2021.
Leonid Kantorovich. On the translocation of masses. J. Math. Sci., 133(4):1381-1382 2006.
Young-geun Kim, Kyungbok Lee, and Myunghee Cho Paik. Conditional wasserstein generator. IEEE Trans. Pattern Anal. Mach. Intell,, 45(6))7208-7219, 2022.
Diederik Kingma and Jimmy Ba. Adam: method for stochastic optimization. In Proc. Int. Conf. Learn. Represent, pages 1-9, 2015.
Dilfira Kudrat, Zongxia Xie, Yanru Sun, Tianyu Jía, and Qinghua Hu. Patch-wise structural loss for time series forecasting. In Proc. Int. Conf. Mach. Learn., 2025.
Henning Lange, Steven L Brunton, and Nathan Kutz. From fourier koopman: Spectral methods for long-term time series prediction. Journal of Machine Learning Research, 22(41):1-38, 2021.
Vincent Le Guen and Nicolas Thome. Shape and time distortion loss for training deep time series forecasting models. In Proc. Adv. Neural Inf. Process. Syst., volume 32, 2019.
Haoxuan Li, Kunhan Wu, Chunyuan Zheng, Yanghao Xiao, Hao Wang. Zhi Geng, Fuli Feng, Xiangnan He, and Peng Wu. Removing hidden confounding in recommendation: unified multi-task learning approach. Proc. Adv. Neural Inf. Process. Syst. 36:54614444444 2024a.
Haoxuan Li, Chunyuan Zheng, Shuyi Wang, Kunhan Wu, Eric Wang, Peng Wu, Zhi Geng, Xu Chen, and Xiao-Hua Zhou. Relaxing the accurate imputation assumption in doubly robust learning for debiased collaborative filtering. In Proc. Int. Conf. Mach. Learn., volume 235, pages 2024b.
Haoxuan Li. Chunyuan Zheng. Wenjie Wang, Hao Wang. Fuli Feng, and Xiao-Hua Zhou Debiased recommendation with noisy feedback. In Proc. ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, page 1576 1586, 2024c.
Jianxin Li, Xiong Hui, and Wancai Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proc. AAAI Conf. Artif. Intell., 2021.
Junjie Li, Yang Liu, Weiqing Liu, Shikai Fang, Lewen Wang, Chang Xu, and Jiang Bian. Mars: a financial market simulation engine powered by generative foundation model. In Proc. Int. Conf. Learn. Represent., 2025a.
Qi Li, Zhenyu Zhang, Lei Yao, Zhaoxia Li, Tianyi Zhong, and Yong Zhang. Diffusion--aase decou- pled deterministic and uncertain framework for probabilistic multivariate time series forecasting. In Proc. Int. Conf. Learn. Represent., 2025b.
Xinyu Li, Yuchen Luo, Hao Wang, Haoxuan Li, Liuhua Peng, Feng Liu, Yandong Guo, Kun Zhang, and Mingming Gong. Towards accurate time series forecasting via implicit decoding. Proc. Adv. Neural Inf. Process. Syst., 2025c.
Shengsheng Lin, Weiwei Lin, Xinyi Hu, Wentai Wu, Ruichao Mo, and Haocheng Zhong. Cyclenet: Enhancing time series forecasting through modeling periodic patterns. In Proc. Adv. Neural Inf. Process. Syst., volume 37, pages 106315-106345, 2024.
Shengsheng Lin, Haojun Chen, Haijie Wu, Chunyun Qiu, and Weiwei Lin. Temporal query network for efficient multivariate time series forecasting. In Proc. Int. Conf. Mach. Learn., 2025.
Minhao Liu, Ailing Zeng, Muxi Chen, Zhijian Xu, Qiuxia Lai, Lingna Ma, and Qiang Xu. Scinet: time series modeling and forecasting with sample convolution and interaction. In Proc. Ady. Neural Inf. Process. Syst., 2022.
11

---

Published as a conference paper at ICLR 2026
Peiyuan Liu, Beiliang Wu, Yifan Hu, Naiqi Li, Tao Dai, Jigang Bao, and Shu-tao Xia. Timebridge: Non-stationarity matters for long-term time series forecasting. In Proc. Int. Conf. Mach. Learn., 2025.
Yong Liu, Tengge Hu, Haoran Zhang, Haixu Wu, Shiyu Wang, Lintao Ma, and Mingsheng Long. itransformer: Inverted transformers are effective for time series forecasting. In Proc. Int. Conf. Learn. Represent., 2024.
Donghao Luo and Xue Wang. Modernten: A modern pure convolution structure for general time series analysis. Proc. Int. Conf. Learn. Represent., pages 1-43, 2024.
Simone Di Marino and Augusto Gerolin. An optimal transport approach for the schrödinger bridge problem and convergence of sinkhorn algorithm. J. Sci. Comput., 85(2):27, 2020.
Yuqi Nie, Nam H Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64
words: Long-term forecasting with transformers. In Proc. Int. Conf. Learn. Represent. 2023.
Gabriel Peyré and Marco Cuturi. Computational optimal transport. Found. Trends Mach. Learn., 11
(5-6):355-607, 2019.
Xihao Piao, Zheng Chen, Taichi Murayama, Yasuko Matsubara, and Yasushi Sakurai. Fredformer: Frequency debiased transformer for time series forecasting. In Proc. ACM SIGKDD Inf. Conf. Knowl. Discovery Data Mining, pages 2400-2410, 2024.
Xiangfei Qiu, Jilin Hu, Lekui Zhou, Xingjian Wu, Junyang Du, Buang Zhang, Chenjuan Guo, Aoying Zhou, Christian S. Jensen, Zhenli Sheng, and Bin Yang. Tfb: Towards comprehensive and fair benchmarking of time series forecasting methods. In Proc. VLDB Endow. pages 2363-2377, 2024.
Xiangfei Qiu, Xingjian Wu, Hanyin Cheng, Xvyuan Liu, Chenjuan Guo, Jilin Hu, and Bin Yang. Dbloss: eecomposition-baseed loss function for time series forecasting. Proc. Adv. Neural Inf. Process. Syst., 2025a.
Xiangfei Qiu, Xingjian Wu, Yan Lin, Chenjuan Guo, Jilin Hu, and Bin Yang. Duct: Dual clustering enhanced multivariate time series forecasting. In Proc. ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, pages 188--199, 2025b.
Hiroaki Sakoe and Seibi Chiba. Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. Signal Process. 26 (6(1):43-49, 2003.
Hao Wang, Zhichao Chen, Jiajun Fan, Haoxuan Li, Tianqiao Liu, Weiming Liu, Quanyu Dai, Yichao Wang, Zhenhua Dong, and Ruiming Tang. Optimal transport for treatment effect estimation. In Proc. Adv. Neural Inf. Process. Syst., volume 36, pages 5404-5418, 2023a.
Hao Wang, Zhichao Chen, Zhaoran Liu, Xu Chen, Haoxuan Li, and Zhouchen Lin. Proximity matters: Local proximity enhanced balancing for treatment effect estimation. Proc. ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2025a.
Hao Wang, Zhichao Chen, Zhaoran Liu, Haozhe Li, Degui Yang, Xinggao Liu, and Haoxuan Li. Entire space counterfactual learning for reliable content recommendations. IEEE Trans. Inf. Forensics Security, 20: 156-1764, 2025b.
Hao Wang, Zhichao Chen, Yuan Shen, Hui Zheng, Degui Yang, Dangjun Zhao, and Buge Liang. Robust missing value imputation with proximal optimal transport for low-quality iiot data. IEEE Trans. Neural Netw. Learn. Syst. 2025c.
Hao Wang, Zhichao Chen, Honglei Zhang, Zhengnan Li, Licheng Pan, Haoxuan Li, and Mingming Gong. Debiased recommendation via wasserstein causal balancing. ACM Trans. Inf. Syst., 2025d.
Hao Wang, Xinggao Liu, Zhaoran Liu, Haozhe Li, Yilin Liao, Yuxin Huang, and Zhichao Chen. Lspt-d: Local similarity preserved transport for direct industrial data imputation. IEEE Trans. Autom. Sci. Eng., 22:9438 048889944,,44 2025e.
Hao Wang, Licheng Pan, Zhichao Chen, Xu Chen, Qingyang Dai, Lei Wang, Haoxuan Li, and Zhouchen Lin. Time-ol: Time-series forecasting needs transformed label alignment. Proc. Adv. Neural Inf. Process. Syst., 2025f.
12

---

Published as a conference paper at ICLR 2026
Hao Wang, Licheng Pan, Yuan Shen, Zhichao Chen, Degui Yang. Yifei Yang, Sen Zhang, Xinggao Liu, Haoxuan Li, and Dacheng Tao. Fredf: Learning to forecast in the frequency domain. In Proc. Int. Conf. Learn. Represent. pages 1 1-9, 2025g.
Huiqiang Wang, Jian Peng, Feihu Huang, Jince Wang, Junhui Chen, and Yifei Xiao. Mien: Multi- scale local and global context modeling for long-term series forecasting, In Proc. Int. Conf. Learn. Represent, 2023b.
Shiyu Wang, Haixu Wu, Xiaoming Shi, Tengge Hu, Huakun Luo, Lintao Ma, James Y Zhang, and Jun Zhou. Timemixer: Decomposable multiscale mixing for time series forecasting. In Proc. Int. Conf. Learn. Represent., 2024.
Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer; Decomposition transformers with CotteCoorelltion for long-term series forecasting. In Proc. Adv. Neural Inf. Process. Syst., 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 Proc. Int. Conf. Learn. Represent. 2023.
Xingjian Wu, Xiangfei Qiu, Hanyin Cheng, Zhengyu Li, Jilin Hu, Chenjuan Guo, and Bin Yang. Enhancing time series forecasting through selective representation spaces: A patch perspective. In Proc. Adv. Neural Inf. Process. Syst., 2025.
Hongteng Xu, Dixin Luo, and Lawrence Carin. Scalable gromov-wasserstein learning for graph partitioning and matching. Advances in neural information processing systems, 32, 2019.
Jingjig Xu, Hao Zhou, Chun Gan, Zaixiang Zheng, and Lei Li. Vocabulary learning via optimal transport for neural machine translation. ACI/IJCNLP (1), pages 7361-7373. Association for Computational Linguistics, 2021.
Kun Yi, Qi Zhang, Wei Fan, Hui He, Liang Hu, Pengyang Wang, Ning An, Longbing Cao, and Zhendong Niu. Fouriergnn: Rethinking multivariate time series forecasting from pure graph perspective. In Proc. Adv. Neural Inf. Process. Syst. 2023a.
Kun Yi, Qi Zhang, Wei Fan, Shoujin Wang, Pengyang Wang, Hui He, Ning An, Defu Lian, Longbing Cao. and Zhendong Niu. Frequency-domat mips are more effective learners in time series forecasting. In Proc. Adv. Neural Inf. Process. Syst., 2023b.
Kun Yi, Qi Zhang, Wei Fan, Longbing Cao, Shoujin Wang, Hui He, Guodong Long, Liang Hu, Qingsong Wen, and Hui Xiong. A survey on deep learning based time series analysis with frequency transformation. Proc. ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, pages 6206-6215, 2025.
Wenzhen Yue, Yong Liu, Haoxuan Li, Hao Wang, Xianghua Ying, Ruohao Guo, Bowei Xing, and Ji Shi. Olinear: A linear model for time series forecasting in orthogonally transformed domain. Proc. Adv. Neural Inf. Process. Syst., 2025.
Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? In Proc, AAAI Conf. Artif. Intell., 2023.
13

---

Published as a conference paper at ICLR 2026
A THEORETICAL JUSTIFICATION
Theorem A.1 (Autocorrelation bias, Theorem 3.1 in the main text). Suppose Yix E R² is the label sequence given historical sequence X, Yix E R is the forecast sequence, Σ|x E RTXT is the conditional covariance of Xjx. The bias of MSE from the negative log- log-likelihood of the label sequence given X expressed as:
|Mx
M 8ix
Bias
where |10112_ -x v 2 v. It vanishes if the conditional covariance 2i is identity matrix'.
Proof. The proof follows the narrative in Wang al. (2025f) but highlights that it is the conditional distribution of Y given X that obeys Gaussian distribution, instead of the marginal distribution of Y.
Suppose the label sequence given X follows a multivariate normal distribution with mean vector Fix [Y ..., Tx and covariance matrix Σ x. The conditional likelihood Y is:
Py|x (2 10.5 exp( 1
11x 8/x 1
(8)
On the basis, the conditional negative log- aggliiklihood of Y is:
log Py|x Tlog((T)) + log |X Mix Yxx 12
Removing the terms unrelated to fix, the terms used for updating f/x, namely practical negative log-likelihood (PNLL), is expressed as follows:
PNLL Mix
Vix
(9)
On the other hand, the MSE loss can be expressed as:
| 8ix|
MSF
(10)
The difference between PNLL and MSE is computed as:
Bias = Vix
9/x |
|ix
81x
(11)
1
which diminishes to zero if the label sequence is conditionally decorrelated, i.e. Σxx is identity matrix. The proof completed.
Lemma A.2 (Lemma 3.3 in the main text). For any p 2 1, the joint-distribution Wasserstein discrepancy upper bounds the expected manditionl-ddistiibnn Wasserstein discrepancy:
f (()[)Ix)P)x)dP(X)) WWP.Pxx,PP..
(12)
where the equality holds pf = 1 or the conditional Wasserstein term is constant with respect to X.
Proof. The proof can be found in Theorem of Kim et al. (2022).
Theorem A.3 (Alignment property, Theorem 3.4 in the main text). The conditional distributions are aligned, i.e., PyІx = P x if the joint distridistiibution Wasserstein discrepancy is minimized to zero, i.e.. (P x,y, xx 0.
'The pioneering work (Wang al., 2025g) identifies the bias under the first-order Markov assumption the lahel sequence. This study generalizes this bias without the first-order Markov assumption.
14

---

Published as a conference paper at ICLR 2026
Proof. By Lemma 3.3, we have
f (p(PPxxx FIx) x( = (,
Thus, if RHS O. have J W, rix, Po 9ix) dP(X) O. Since Wp is non-negative (Peyré and Cuturi, 2019), this implies that , (1x, Py p(xx) = for almost every X. Therefore, suffices to prove that for two distributions P.. = pyxx and P3 P (P, P3) = 0 implies P., Pp.
Suppose S. (00,, On] and S3 [,,, 3n] the empirical samples from P.. and Pa. respectively, with corresponding mass vectors a and We are given that (IP., Pp)) = 0. By Definition 3.2, this means the minimum value of the cost function is zero. Let p be an optimal transport plan that solves the minimization problem. Then,
m
(PP P3) (D, P*)
Σ
Piy ||;
Bylly O.
(13)
From the constraints, we know the elements of the transport plan are non-negative, Př, 0. The distance term is also non-negative, l|a; - 3 2 0. Since the total sum these non-negative terms zero, each individual term in the summation must zero:
Pry |la B O, Vi I,., n 1,..,
(14)
This condition implies that if any mass is moved from a point as a point By (i.e., P.j > 0), then the distance between these points must be zero (i.e., ||a, - 3, l| 0), which means a; = By. In other words, the optimal plan only transports mass between identical points.
Let's consider the total probability mass assigned of either distribution. The total mass at 2, i.e., probability density, for distribution Pa is Pa(z) Eia.= ag Using the constraints from II(P... P3) in Equation 3, can express this as:
arbitrary value that exists in the support
m
Pa(=)
Σ 04 Σ c0
Σ
P
(15)
co,=z
As established, P. can only be non-zero if β, 0g Therefore, for the outer sum where o, the inner sum over j is non-zen zezzero only for those indices j where By = 2. Thus, we can write:
(a(=)= Σ E Pis-
(16)
ba;= j:8,=z
Similarly, the mass at z for distribution P. is P3(2) = Ej:8, = Using the other set of constraints from (1 Pa):
by.
Pa(=) = Σ b, Σ
Σ P
(17)
j0 j&8,=
Again, since P",, is non-zero only a; By. for the terms in the outer sum where B3 = 2, the inner sum over is non-zero only for those indices where On; z. This gives:
Pa(2) = Σ Σ P
(18)
8:8;=z k0,=z
By comparing the resulting expressions for Pa (2) and Po(z), find they are identical:
= Pz(z))
(19)
Since this equality holds for any value z, the probability mass functions of P. and P& are identical, which implies P., = 3 Applying this result to our conditional distributions, (Pyix, = implies Fy|x = F yyyx for almost every X. This completes the proof.
A discrete probability is completely characterized by two components: its support and its probability mass function.
15

---

Published as a conference paper at ICLR 2026
Lemma A.4 (Lemma 3.5 in the main text). Suppose Px, and IP x, obey Gaussian distributions Y Ex and x Ex,y)) respectively. The squared W2 discrepancy can be calculated as the Bures- wares-Wassesstein discrepancy:
1 BW Ex,, E -x,x) |XX, - 1x, 11: + B ,
(20)
where B(Ex,,, Ex,y) = Tr Exy + E x, 2 x, E x D 1/2 X.Y
Tr(-) denotes matrix trace.
Proof.
The proof can be found in Remark 2.31 of Peyré and Cuturi (2019).
Additional notes on the Gaussian assumption. Lemma 3.5 presents the BW discrepancy under the Gaussian assumption, yielding a tractable and efficient form. However, the Bures- Warres-Wsseesttein discrepancy measures differences only in the first- and second-order moments-i.e., the mean and covariance. While these two moments fully characterize Gaussian distributions, real-world datasets do not necessarily adhere to Gaussianity, additionally requiring higher-order moments for complete characterization. Nonetheless, the mean and covariance remain essential descriptors for any distribution. As a result, in cases where data deviate from strict Gaussianity, BW remains a valuable tool for distribution alignment by matching these fundamental moments.
B OVERVIEW OF DISCRETE OPTIMAL TRANSPORT AND WASSERSTEIN DISCREPANCY
This section outlines the foundational concepts of optimal transport (OT) and the Wasserstein discrepancy. Our analysis is specifically confined to discrete probability measures, as the broader theory involving general measures is beyond the scope of this work. For comprehensive treatment of the continuous case, readers are directed to the seminal works by Peyré and Cuturi (2019).
The classical framing of OT, known as the Monge problem, can be illustrated with simple scenario: transporting goods from n warehouses to m factories (Peyré and Cuturi, 2019). Let the i-th warchouse hold a, units of material and the j-th factory require by units. The objective is to find transport map that moves all material from the warehouses to satisfy the factories' demands. This problem is subject to several constraints: the entire stock from each warehouse must be shipped, all factory demands must be met, and the mapping must be deterministic (i.c., each warehouse ships its entire stock to a single factory). The optimal map is the one that minimizes the total cost, which is aggregated from the cost of moving a unit of material from a given warehouse to a factory.
Definition B.1 (Monge Problem for Discrete Measures). Let Ei= 0x,x, and Σド= bydy, in be two discrete probability measures. The Monge problem seeks a transport map T { +
(y that pushes the mass of a forward to match B, denoted by T,a = B. This condition implies that for each j. the total mass received, by, must equal the sum of the masses sent from all locations mapped to it: bj- EiT()))= 0g. The objective is to find the map T that minimizes the total transportation cost:
min T-Ta==
c(x;, I(x;))a
(21)
While intuitive, the Monge formulation is restrictive; a solution is not guaranteed to exist, particularly when mass splitting is required (e.g., one warehouse supplying multiple factories). To address this limitation, Kantorovich (2006) introduced a relaxed formulation. Instead of deterministic map, Kantorovich's 's approach seeks a probabilistic coupling or "transport plan" that allows mass from single source to be distributed among multiple destinations. This reframes the problem within the versatile framework of linear programming. When the measures are probability distributions (i.e., Ea,- b; = 1), the resulting optimal cost defines a distance metric. Definition B.2 (Kantorovich Problem). Let a
Σn= agdx, and B = =T byöy, be two discrete probability distributions supported on samples {x and {y respectively. The optimal transport problem is to find a transport plan π E R that minimizes the total cost:
Wa(o, == min (C, T));
(22)
=Ell(a,6)
16

---

Published as a conference paper at ICLR 2026
Table 7:
Dataset description.
Dataset
D
Forecast length
Train validation / test
Frequency
Domain
ETTh1
96, 192, 336, 720
8544//281/2881
Hourly
Health
ETTh2
96, 192, 336, 720
8545/2881/2881
Hourly
Health
ETTml
7
96, 192, 336, 720
34465/11521/11521
15min
Health
ETTm2
7
96, 192, 336, 720
34465/11521/11511
15min
Health
Weather
21
96, 192, 336, 720
36792/5271/105400
10min
Weather
ECL
321
96. 192. 336, 720
18317/2633/5261
Hourly
Electricity
Traffic
862
96, 192, 336, 720
12185/1757/3509
Hourly
Transportation
PEMS03
358
12, 24, 36, 48
15617/5135/5135
5min
Transportation
PEMS08
170
12, 24, 36, 48
10690/3548/265
Smin
Transportation
Note: D denotes the number variates. Frequency denates the sampling interval of time points. Train, Volidation, Test denotes the number of samples employed each split. taxonomy aligns with (Wo et al., 2023).
where (F is the Frobenius dot product. The cost matrix C E IR contains the pairwise costs, egg,, Cig V;). The set of feasible transport plans, I1 b). is defined by the constraints that preserve the total mass of the source and target measures:
II(a, b) == { E R4
T1m a, π b).
(23)
Here, and b are the weight vectors for the measures a and B. If the cost is a metric distance raised a power p. y(x,y)= |xx y yl,,, the p-th root of the optimal cost defines the p- --aassesttein discrepancy, B).
Contemporary research in discrete optimal transport primarily progresses along two paths. The first focuses on computational efficiency. Exact solutions via linear programming are often infeasible for large-scale problems due to their high computational complexity, typically O( log n) where is the number of support points (Bonneel et al., 2011). This has motivated the development of faster, approximate methods, such as entropic regularization (leading to the Sinkhorn algorithm) with nearly quadratic complexity (Altschuler et al., 2017) and sliced OT, which reduces the problem to one- diee-imennional computations and achieves near-linear complexity. The second path involves adapting the OT framework address specific challenges across various domains, such domain adaptation (Chizat et al., 2018), causal inference (Wang et al., 2025a; 2023a), generative modeling (Marino and Gerolin, 2020; Chen et al., 2024), missing data imputation (Wang et al., 2025e;c), graph comparison (Xu et al., 2019) and recommendation system (Wang et al., 2025d).
C REPRODUCTION DETAILS
C.1 DATASET DESCRIPTIONS
Our empirical evaluation is conducted diverse collection of widely-used time series forecasting benchmarks. Each dataset presents distinct characteristics in terms of dimensionality and temporal resolution. A summary is provided in Table 7.
ETT (Li et al., 2021): Contains seven metrics related to electricity transformers, recorded from July 2016 to July 2018. It divided into four subsets based on sampling frequency: ETTh and ETTh2 (hourly), and ETTml and ETTm2 (every 15 minutes).
Weather (Wu et al., 2021): Comprises 21 metcorological variables from the Max Planck Biogeo-
chemistry Institute's weather station, captured every 10 minutes throughout 2020.
ECL (Wu et al., 2021): Features the hourly electricity consumption of 321 clients.
Traffic (Wu et al., 2021): Documents the hourly occupancy rates 862 sensors on San Francisco
Bay Area freeways, spanning from 2015 to 2016.
PEMS (Liu et al., 2022): Consists of public traffic data from the California highway system, aggregated in 5-minute intervals. We utilize two common subsets, PEMS03 and PEMS08.
17

---

Published as a conference paper at ICLR 2026
Following established protocols (Qiu et al., 2024; Liu et al., 2024), all datasets chronologically partitioned into training, validation, and test sets. For the ETT. Weather, ECL, and Traffic datasets, we fixed historical sequence length of 96 and evaluate performance across four prediction horizons with lengths of 96, 192, 336, and 720. For the PEMS datasets, we also use an historical length of 96 but evaluate shorter prediction horizons of 12, 24, 36, and 48 steps. During the final evaluation on the test set, we ensure that no data is discarded from the last batch: a technique referred to as the dropping-lat last trick is disabled throughout our experiments.
C.2 IMPLEMENTATION DETAILS OF MODEL TRAINING
To establish fair comparison, we reproduced all baseline models using their official, publicly available implementations, primarily sourcing from the iTransformer (Liu et al., 2024) and Fred- former (Piao et al., 2024) repositories. The reproducibility of these bascline results was verified prior to our experiments. All models were trained to minimize the MSE loss function using the Adam optimizer (Kingma and Ba, 2015). The learning rate for each baseline selected from the set {10-3,5 x 10-2, 10-1,1 5 x 10-5} based on the best performance on the validation set. To prevent overfitting, we employed an early stopping mechanism that terminates training if the validation loss fails to improve for three consecutive epochs.
When integrating our proposed distributional discrepancy component, DistDF, with an existing forecasting model, we maintain the original model's ' optimized hyperparameters as reported in their respective benchmarks (Liu al., 2024; Piao al., 2024). Our tuning therefore focused and conservative, limited to two key parameters: the learning rate and the weight of the discrepancy term, (0, 1]. Adjusting the learning rate is necessary as the distributional discrepancy term has varying overall magnitude and gradient dynamics on different datasets. The tuning is driven by selecting the combination that yields the lowest MSE on the validation set.
C.3 IMPLEMENTATION DETAILS OF CONDITIONAL CORRELATION COMPUTATION
key challenge in analyzing time series is accurately quantify the autocorrelation structure within the label sequence without the confounding influence of the historical sequence Wang et al. (2025b); Li al. (2024b). Standard metrics like the Pearson correlation insufficient for this task, as they cannot disentangle the dependencies among future time steps from their shared dependence on the past Li al. (2024a;c).
To address this, we employ the partial correlation coefficient to measure the conditional autocorre- lation. This allows to assess the relationship between any two time steps in the label sequence while controlling for the linear effects of the entire historical sequence. Our implementation is based on the standard procedure for computing partial correlation, which also implemented in statistical software like MATLAB's 'partialcorr' function."
The procedure can be described as follows. Let X the historical sequence (the control variables) and Y be the label sequence. To compute the partial correlation between two time steps, Y, and Ye, conditioned on X, we follow a two-stage regression process. We first isolate the variance in Y, and Yv that cannot be explained by X. This is achieved by training two separate linear regression models using ordinary least squares (ÖLS). The residuals from these models, e and ét,, represent the parts of Y, and Y, that linearly independent of X. The partial correlation between Y, and Yr, conditioned on X, is then calculated as the standard Pearson correlation between their respective residuals. This process effectively measures the linear relationship between Y, and Yp after accounting for the influence of the historical context X.
D MORE EXPERIMENTAL RESULTS
D.1 OVERALL PERFORMANCE
Additional experimental results of overall performance are available in Table 8, where the performance given different T is reported.
"mmllementation is available at
html
18

---

Published as a conference paper at ICLR 2026
Table 8: Full results on the multi-step forecasting task. The length of history window is set 96 for all baselines. Avg indicates the results averaged over forecasting lengths: T=96, 192, 336 and 720.
DerDF (Curs)
TineBridge (2025)
Fredlormer (2004)
iTranadocusr (2024)
FreTS (2023)
TimesNet (2023)
MICN (3023)
TIDE
PaicTTT (2023)
DLinear (2025)
Model
(2023)
Meuics
MSE
MAE
MSE
MAE
MSE
MAE
MAE
MSE
MAE
MSE
MAE
MSB
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0,316
0.357
0.323
0.361
0.326
0.361
0.338
0.572
0.342
0.375
0.368
0.394
0.319
0,365
0.353
0.374
0.325
0.364
0.346
0,313
192
0.365
0.350
0.366
0.365
0.362
0.506
0.385
0.400
0.406
0.409
0.364
0.395
0.391
0.303
0.363
0.383
0.580
0.390
3366
0,392
0.404
0.908
0,408
0.196
0.404
0.427
0.434
0.416
0.421
0.454
0.444
1.955
0,425
0.423
0.414
0.404
0.413
0.43
0,414
720
0.485
0.437
0.461
0,445
0.459
0.464
0.496
0.463
0.513
0.489
0,527
0.414
0.505
0.499
0.485
0.448
0.463
0.642
0.472
0,450
Avrg
00,77%
0.994
0.387
0.400
0.182
0. 198
0.411
0.414
0.414
0.421
0.438
0.430
0.906
0.421
0.41%
0.407
0.389
0.400
0.405
0.400
96
0.374
0.256
0.177
0.350
0.177
0.260
0.182
0.265
0.188
0.279
0.184
0.262
0.178
0.277
0.182
0.265
0.190
0.366
0.188
0.283
ETTh2
192
0.289
0.298
0.243
0.903
0.342
0.
0.315
0.364
0.929
0.257
0.988
0.266
0.343
0:347
0.904
0.265
0.389
0.280
0.356
336
0,300
0.338
0.300
0.340
0.356
0.309
0,315
0,545
0.299
0.354
0.307
0.345
0.309
0.547
0.575
0,420
720
0.397
0.394
0.401
0.399
0.389
0.392
0.423
0.411
0.489
0.482
0.452
0.421
0.459
0.482
0.405
0.398
0.437
0.422
0.526
0,508
Avg
0.277
0.321
0.251
0.326
0.280
0.324
0.295
0.536
0.316
0.365
0.302
0.334
0.308
0.364
0,286
0.328
0.303
0.344
0.342
0.392
96
0,373
0.393
0.313
0.399
0.317
0.396
0.385
0.405
0.398
0.409
0.399
0.418
0.391
0.415
0.387
0.395
0.381
0.400
0.389
0.404
ETTbJ
192
0:425
0.425
0.118
0/426
0,437
0.425
0.440
0.637
0.451
0.443
0.453
0.851
0.897
0.489
0.439
0.425
0:450
0,643
0.642
0,440
336
0,466
0.457
0.472
0,488
0.009
0.589
0.555
0.482
0.447
0.501
0.420
0.488
0.467
720
0.453
0.453
0.455
0,487
0,488
0.462
0.504
0.492
0.608
0.571
0.549
0.515
0.665
0.517
0.484
0.471
0.504
0.492
0.905
0.502
Avg
01:30
0.429
0.442
0.440
0.447
0.454
0.452
0.448
0.489
0.474
0.472
0.468
0,533
0,519
0.448
0.485
0:499
0:451
0.456
0.453
96
0,287
0.336
0.294
0.344
0.293
0.344
0.301
0.349
0.315
0.374
0.321
0.358
0.351
0.398
0.201
0.340
0.299
0.349
0,330
0.383
E1T12
192
0,358
0.351
0.311
0.394
0.372
0.391
0.383
0.997
0.406
0.467
0.418
0.417
0.892
0.489
0.376
0.392
0.383
0,404
0.439
0,450
336
0.432
0.502
0,464
0.454
0.656
0.582
0.417
0.439
0.444
0.589
0.538
720
0.416
0.435
0.423
0.443
0.421
0.450
0.436
0.448
0.702
0.643
0.434
0.450
0.981
0.718
0.420
0.446
0:438
0.455
0.757
0.626
Avr.
0,367
0.398
0.377
0.403
0.377
0.402
0.386
0.407
0.524
0.496
0,409
0.420
0.620
0,546
0.378
0.401
0.390
0,415
0.529
0.499
96
0,137
0.235
0.142
0.330
0.161
0.258
0.150
0.242
0.180
0.266
0.170
0.212
0.170
0.281
0.197
0.274
0.170
0.364
0.197
0.282
192
0.189
0.257
0.161
0.257
0.124
0.268
0.168
0.259
0.186
0:272
0.181
0.262
0.185
0.297
0.197
0.217
0.119
0.215
0.197
0.786
BCL
336
0.178
0.274
0.203
0.302
0.190
0.298
0.312
0.292
0.195
0.288
0.209
720
0.212
0.962
02217
0.909
0.235
0.319
0.214
0. 996
0.214
0.323
0.794
0.966
0.231
0.329
0:384
0.325
0.254
0,300
0.345
0.334
Avg
0.173
0.367
0.176
0.271
0.191
0.264
0.179
0.270
0.199
0,288
0,212
0.506
0.192
0,302
0.315
0.292
0.195
0,286
0,212
0,301
96
0.380
0.262
0.901
0.368
0.461
0.327
0.397
0.271
0.511
0.323
0.590
0.316
0.498
0.298
0.646
0.986
0.444
0.284
0.649
0.390
Traffic
192
0,407
0.275
0.418
0.276
0,470
0.326
0.416
0.279
0.519
0.321
0.624
0.536
0.521
0.309
0.599
0.362
0.454
0.291
0,598
0.371
336
0.539
0.606
D:460
720
0.453
0.297
0.964
0.301
0,521
0.353
0.462
0.305
0.575
0,345
0,670
0.356
0.567
0.326
0.643
0.383
0.500
0.319
0.646
0,395
AAg
0.417
0.279
0.426
0.282
0.486
0.336
0.426
0.285
0.538
0.330
0.631
0.338
0.529
0,312
0.624
0.373
0.468
0.298
0.625
0,364
96
0.164
0.209
0.108
0,180
0.171
0.210
0.174
0.228
0.183
0.239
0.179
0.244
0.192
0.252
0.189
0,230
0.194
0.253
Weather
192
0,212
0.252
0.214
0.254
0.222
0.258
0.246
0.278
0.213
0.266
0.241
0.276
0.242
0.310
0.240
0.270
0.228
0.202
0.238
0.296
3366
0,270
0,203
0.312
0.273
0.330
0.292
0.307
0.268
0.305
0.292
0.332
720
0,345
0.345
0,353
0.347
0.358
0.348
0.362
0.355
9.337
0,362
0.366
0.361
0.350
0.399
0.364
0.353
0.362
0.354
0.547
0,365
Avg
0,245
0.275
0.252
0.377
0.361
0.262
0.260
0.289
0.240
0.293
0.271
0.205
0.264
0.321
0.272
0.201
0.267
0.288
0.265
0.317
12
0.065
0.174
0.070
0.179
0.085
0.198
0,094
0.201
0.096
0.217
0.117
0.226
0.092
0.210
0.105
0.220
PEMSU3
24
0,894
0.205
0.009
0.211
0.121
0.240
0.104
0.217
0.129
0.244
0.116
0.221
0.095
0.210
0.233
0.322
0.144
0.263
0.183
0.297
36
0.251
0.194
0.217
0.107
0.379
0.500
0.909
0.258
48
0.338
0.252
0.153
0.267
0.201
0.316
0.174
0.285
0.207
0.315
0.161
0.262
0.125
0.242
0.535
0.516
0.245
0.344
0.319
0,410
Avg
0.104
0.215
0.122
0.223
0.146
0.260
0.122
0.233
0.149
0.261
0.126
0.210
0.106
0.223
0.316
0.370
0.170
0.282
0.216
0.322
12
0.076
0.000
0.187
0.096
0.203
0.111
0.208
0.161
0.274
0.121
0.233
0.106
0.223
0.113
0.225
PEMSUS
24
0.107
0.200
0.119
0.334
0.138
0.245
0.123
0.227
0.151
0.258
0.1.9
0.212
0.127
0.237
0.382
0.325
0.162
0.225
0.199
0.302
36
0.268
0.300
0.168
0.280
0.148
0.252
0.376
0.427
0.254
0.295
48
0,371
0.365
0.198
0.289
0.255
0.338
0.218
0.306
0.247
0.354
0,189
0.272
3.175
0.270
0.543
0.527
0.301
0,382
0,389
0.429
Axg
0.123
0.233
0.139
0,239
0.121
0.271
0.149
0.247
0.134
0.275
0.152
0.243
0.153
0.258
0.318
0.318
0.201
0,305
0.249
0.332
Count
0
D.2
SHOWCASE
Additional experimental results of showcases are available in Fig. 4 and Fig. 5, where two datasets and two forecast models are involved.
D.3 COMPARISON WITH DIFFERENT LEARNING OBJECTIVES
Additional experimental results of learning objective comparison are available in Table 9, where two forecast models evaluated across different T values.
19

---

Published as a conference paper at ICLR 2026
-0.5
-1.0
103
230
500
403
100
200
500
400
100
200
900
400
100
200
800
400
(a) Snapshot with TimeBridge
(b) Snapshot
Fredformer
WW
-3.5
.2.
-55
-05
-1.0
-10
-10
-1.5
GroundTruth
DrounsTruh Pledicaion
-15
Groundtiuuh Find crtion
GroundTuhh
Praciation
Prodiction
-20
-20
-20
-20
(c) Snapshot 2 with TimeBridge
(d) Snapshot with Fredformer
M
W
-05
-10
-30
-10
-1.5
103
200
603
200
603
200
400
200
(c) Snapshot 3 with TimeBridge
(f) Snapshot 3 with Fredformer
Figure 4: The forecast sequences generated with DF and DistDE. The forecast length set to 336 and the experiment is conducted ETTm2.
TrpandTrath Preceatin
GrapanSrruhh Precidtion
GrouuddTrshh Prrict
-08
-0.8
-0.8
-.88
100
200
200
100
200
2000
100
200
200
100
200
300
(a) Snapshot with TimeBridge
(b) Snapshot
Fredformer
0.0
0.0
Prodiotion
Ryedictiion
Prediction
Predittion
(c) Snapshot 2 with TimeBridge
(d) Snapshot with Fredformer
Grount
GroundTiuh
MM MoW
M
303
100
200
802
100
200
300
100
230
300
(c) Snapshot with TimeBridge
(f)
Snapshot with Fredformer
Figure S: The forecast sequences generated with DF and DistDE. The forecast length and the experiment is conducted on ECL.
192
20

---

Published as a conference paper at ICLR 2026
Table 9: Comparable results with different learning objectives.
Loss
DisIDF
Time-ol
FreDF
Koopman
Dilate
Soft-DTW
DE
Metries
MAF
MSE
MSF
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Forecast model:
Timmeridge
96
0.319
0.358
0
0.356
0.325
0.361
0.572
0.493
0.321
0.360
0.321
0.359
0.323
0.361
E77ml
192
0.363
0.383
0.363
0.382
0.373
0.385
0.410
0.407
0.366
0.386
0,368
0.385
0.366
0,385
336
0.394
0.405
0.396
0.407
0.398
0.406
0.397
0.408
0.397
0.409
0.405
0.410
0.398
0.408
720
0.455
0.442
0.456
0.443
0,450
0.438
0.460
0.445
0.462
0.447
0.486
0.453
0.461
0.445
Avg
0,383
0.397
0.383
0,397
0.386
0.398
0.460
0.438
0.387
0.400
0.395
0.402
0.387
0.400
96
0,372
0,392
0.372
0
0.373
0.391
0.376
0.397
0,376
0.396
0.376
0.395
0.373
0.395
ETThl
192
0.424
0.429
0.422
0.423
0.425
0.421
0.426
0.430
0.430
0.435
0,425
0.427
0,428
0,426
336
0.467
0.450
0.468
0,450
0.467
0.442
0.483
0.461
0.49%
0.469
0.481
0.458
0.471
0.451
720
0.472
0.471
0.495
0.488
0.493
0.490
0.551
0.509
0.552
0.509
0.529
0.499
0.495
0.487
Avg
0.434
0.436
0.439
0.438
0.439
0.446
0.459
0.449
0.464
0.452
0.452
0.445
0.442
0.440
96
0.137
0,235
0,148
0,240
0.137
0.232
0.170
0.266
0.142
0.240
0.139
0.235
0,142
0.239
192
0.159
0.257
0.156
0.251
0.159
0.254
0.161
0.258
0.160
0.257
0,160
0.257
0.161
0,257
BCL
336
0,178
0.272
0.177
0,273
0.179
0.273
0.182
0.277
0.182
0.277
0.178
0.274
0,182
0.278
720
0.212
0.302
0.220
0.308
0.224
0.310
0.217
0.308
0.218
0.309
0.215
0.305
0.217
0.309
Avg
0.172
0,267
0.175
0,268
0.175
0.267
0,182
0.277
0.176
0.271
0.173
0.268
0.176
0,271
96
0,164
0,209
0.166
0,209
0.174
0.213
0.215
0.261
0.168
0.211
0.169
0.209
0.168
0.211
Weather
192
0.212
0.252
0.212
0.252
0.223
0.255
0.239
0.271
0.214
0.254
0,215
0.251
0.214
0,254
336
0.270
0.295
0,270
0.294
0.271
0.292
0.271
0.295
0.273
0.297
0.275
0.296
0.273
0.297
720
0.348
0.345
0.352
0.347
0.350
0.346
0.350
0.345
0.353
0.347
0.379
0.364
0,353
0.347
Avg
0,248
0,275
0.250
0.275
0.254
0.276
0.269
0.293
0.252
0,277
0,260
0,280
0,252
0,277
Forecasi modill: FredFormer
0.316
0.357
0.321
0.357
0.326
0.355
0.335
0.368
0.337
0.367
0.332
0.363
0.326
0.361
ETTml
192
0.358
0,380
0.360
0,378
0.363
0.380
0.366
0.384
0.364
0.384
0,370
0.386
0.365
0,382
336
0.392
0.389
0.392
0.400
0.408
0.397
0.406
0.409
0.396
0.404
720
0.448
0.437
0.447
0.435
0.455
0.440
0.456
0.441
0.457
0.443
0.478
0.450
0.459
0.444
Avg
0.378
0.394
0.379
0.393
0.384
0.394
0.389
0.400
0.389
0.400
0.397
0.402
0.387
0.398
0.368
0.370
0.392
0.397
0.378
0.376
0.398
0.396
E7Th
192
0.428
0.425
0.424
0.422
0.436
0.437
0.438
0.434
0.439
0.435
0.439
0.435
0.437
0.425
336
0.466
0.467
0.441
0.473
0.443
0.473
0.455
0.481
0.484
0.455
0.486
0,449
720
0.453
0.453
0.465
0,463
0.474
0.466
0.523
0.487
0,516
0.482
0.542
0.510
0,488
0.467
Avg
0.430
0.429
0.431
0.429
0.438
0.434
0.452
0.443
0.453
0.442
0.460
0.449
0.447
0.434
0.238
0.151
0.152
0.247
0.263
0.158
0.168
0.266
0.258
BCL
192
0.162
0.255
0.166
0.256
0.166
0.257
0.174
0.267
0.170
0.263
0,218
0.313
0.174
0.269
336
0.176
0.270
0.181
0.274
0.183
0.278
0.188
0.280
0.190
0.286
0.197
0.291
0.194
0,290
720
0,211
0,300
0.213
0.304
0.216
0.304
0.232
0.318
0.229
0.316
0.240
0.322
0.235
0.319
Avg
0.173
0.266
0.178
0.270
0.179
0.272
0.190
0.282
0.187
0.280
0.206
0.298
0.191
0.284
0.172
0.212
0.171
0.208
0.174
0.213
0.174
0.214
0.173
0.173
0.213
0.220
Weather
192
0,218
0.255
0.219
0.253
0.219
0.254
0.220
0.256
0.225
0.260
0.220
0.255
0.222
0.258
336
0.277
0.297
0.277
0.295
0.278
0.296
0.280
0.298
0.280
0.299
0,281
0.296
0.283
0.301
720
0.352
0.347
0.353
0.346
0.354
0.347
0.354
0.347
0.355
0.348
0.369
0.355
0.358
0.348
Avg
0.255
0.277
0.255
0.276
0.256
0.277
0.257
0.279
0.258
0.280
0.261
0.280
0.261
0.282
21

---

Published as a conference paper at ICLR 2026
with DedEf We Datte
0.40
VIW with TlxED
0.51
2W5 with DietD
0.55
with a1
Mil DNGL
DidDF
Wil Dhedkrr
0.90
0.39
2:48
TAAE
0.42
D-45
0.36
0.35
0.36
MSE
MSC
9.40
0.36
0.24
Time
Fied.
ITrnss
FroTs
Time
Fred
ITrons
FieT3
Time
Fredi
Iarns.
FreTs
Time.
Fird
iTooss
PieTB
(a) ETTml
(b) ETTm2
(c) ETTh1
(d)
ETTh2
Figure 6: Performance of different forecast models with and without DistDF. The forecast errors are averaged over forecast lengths and the error bars represent 50% confidence intervals.
Table 10: Varying input sequence length results on the Weather dataset.
Models
DistDF
TimeBridge
DistDF
PatchTST
Metrics
MSF
MAE
MSF
MAF
MSF
MAE
MSF
MAF
96
0.164
0.209
0.168
0.211
0.179
0.220
0.189
0,230
0.254
0.257
96
336
0.270
0.295
0,273
0.297
0.278
0.298
0,288
0,305
720
0.348
0.345
0.353
0.347
0.354
0.348
0.362
0.354
Histrical sequence length
Avg
0.248
0.275
0,252
0,277
0.258
0.281
0.267
0,288
96
0.160
0.207
0,163
0,210
0.157
0.203
0.163
0.209
0,248
0.244
192
3366
0.260
0.290
0,259
0,288
0.258
0.285
0.268
0,293
0.335
0.342
0,344
0.338
0,511
0.451
Avg.
0.239
0.271
0.241
0.273
0.238
0.267
0.287
0.301
96
0.155
0.206
0.156
0.206
0.153
0.204
0.158
0.208
0.245
0.249
0.235
0.291
336
336
0.245
0.283
0,259
0.294
0.250
0.285
0,252
0,287
720
0.325
0.337
0.323
0.335
0.323
0.337
0.326
0.336
Avg
0.231
0.267
0.234
0,270
0.232
0.269
0.243
0,280
96
0.147
0.198
0.148
0.201
0.149
0.204
0.153
0.205
0.247
0.254
720
3.36
0.240
0.279
0,239
0,278
0.247
0.291
0.248
0,288
0.319
0.339
0.346
0.333
0.317
0.339
Avg
0.226
0.266
0.230
0.269
0.226
0.269
0.231
0.272
D.4 GENERALIZATION STUDIES
Additional experimental results of varying forecast models are available in Fig. 6, where four forecast models are involved on four datasets.
D.5 CASE STUDY WITH PATCHTST OF VARYING HISTORICAL LENGTHS
Additional experimental results of varying historical lengths available in Table 10, complementing the fixed length of 96 used in the main text. The forecast models selected include TimeBridge (Liu et al., 2025) which is the recent state-of-the-an the-art forecast model. and PatchTST (Nie et al., 2023) which is known to require large historical lengths. The results demonstrate that DistDF consistently improves both forecast models across different historical sequence lengths.
22

---

Published as a conference paper at ICLR 2026
Table 11: Experimental results (stan with varying seeds (2021-2025).
Dataset
ECL
Weather
Models
DistDF
DF
DistDF
DF
MSE
0.138
0.236
00.001
0.239 001
0.167
0.209 400.01 0.001
0.169 001
0 001
0.15940.001
0.257
0.161 =0.000
0,258 001
0.213
0.253+0.000
0.215+0.000
0.254
0.1791100011
0.272.0.001
20.002
0.279 002
0.271
0.296.0..00 0.2
0.272..0.00
0.296. 001
0.21010.000 001
0. 20000
20.005
0.31 004
0.349g0.002
0.3472...002
0.382 0.0
0.348 001
0.172.20.000
0.260=0.0om
0.177.20.022
0.272 001
0,276 0.200
0.252
0.277 000
0.64
0.70
0.74
0.60
(a) Running time in the forward phase.
(b) Running time in the backward phase.
Figure 7: Running time (ms) with varying forecast horizons.
D.6 RANDOM SEED SENSITIVITY
Additional experimental results random seed sensitivity are available in Table 11, where we report the mean and standard deviation of results obtained from experiments conducted with five different random seeds (2021, 2022, 2023, 2024, and 2025). The results indicate minimal sensitivity of the proposed method random initialization, as averaged standard deviations remain below 0.005.
D.7 COMPLEXITY
Additional experimental results of the running time of DistDF are available in Fig. 7. The batch size and dimension are set to 128 and 21, respectively. As the forecast horizon T increases, the running time for both forward and backward passes generally rises, with some fluctuations. This trend is expected, since T affects the size of the matrices involved in computing the joint-distribution Wasserstein discrepancy in (5). Nevertheless, the running time remains below 1 even when T increased to 1024. Furthermore, DistDF's additional computations occur exclusively during training and are completely isolated from the inference stage.
As result, DisDD introduces no additional complexity to model inference, and the extra computa- tional cost during training is negligible.
D.8 JOINT-DISTRIUTTIN DISCREPANCY IN VARYING SETTINGS
Additional experimental results of joint distribution discrepancy available for different learning objectives in Table 12 and a values in Table 13 and Table 14, as supplement to Table 2, Table 5 and Table 6. The joint distribution discrepancy, denoted Disc, is evaluated on the test set to compare the discrepancy between (X, Y) and (X, Y).
D.9 UTILITY TO IMPROVE RECENT FORECASTING MODELS
Additional experimental results demonstrating utility for improving recent forecast architectures are available in Table 15. We select TQNet (Lin et al., 2025), TimeBridge (Liu et al., 2025), and FredFormer (Piao al., 2024) testbeds due to their recency and competitive performance.
23

---

Published as a conference paper at ICLR 2026
Table 12: Joint-distribution discrepancy of different objectives for time-series forecasting.
Loss
DistDF
Time-ol
FreDF
Koopman
Dilate
LDTW
Soft-DTW
DTW
DF
Timeleidge
ETTml
0.230
0.231
0.231
0.271
0.231
0.231
0.238
0.237
0.232
ETThl
0.326
0.331
0.330
0.350
0.352
0.352
0.340
0.344
0.332
ECL
0.129
0.135
0.137
0,139
0.136
0.139
0.133
0.140
0.136
Weather
0.147
0.148
0,149
0.157
0.148
0.148
0.153
0.150
0.148
Frofloomer
ETTml 0.227
0.228
0.231
0,232
0.233
0.233
0.240
0.239
0.232
0.324
0.325
0.349
0.349
0.342
ECL
0.130
0.133
0,134
0,142
0.140
0.144
0.153
0.151
0.143
Weather
0.148
0.148
0.149
0.150
0.150
0.152
0.152
0.152
0.152
sccond-hest jesnlts rospectiolly. The reported results everagol over Fornosst horizons: T-96, 192, und 220. When metric valuess voincide to three decimal places, Bold indicatiss the numerically superior result based on full precision..
underlined
Table 13: Varying a results where Timebridge acts as the forecasting model.
ETTh2
Weather
a
MAE
Disc
MSE
MAE
0.377
0.403
0.292
0.176
0.271
0.1.6
0,252
0.277
0.148
0.001
0.378
0,402
0.292
0.172
0,267
0.130
0.250
0.276
0.148
0.002
0.377
0.402
0.291
0.173
0.267
0.130
0,250
0.276
0,148
0.376
0.291
0,250
0.01
0.376
0.400
0.291
0.172
0,267
0,130
0.249
0.276
0.146
0.02
0.376
0,400
0.291
0.174
0.269
0.133
0,249
0.276
0.147
0.05
0.375
0,399
0.290
0.174
0,268
0.132
0.251
0.278
0,147
0.1
0.375
0.399
0.291
0.174
0.269
0.132
0.254
0.280
0. 14448
0.376
0,399
0.291
0.177
0,270
0.134
0.254
0.280
0.148
0.5
0.378
0.400
0.294
0.186
0.277
0.140
0,256
0.281
0,149
0.381
0.402
0.296
0.197
0.147
0,260
0.283
0.150
Note: undertined superior result based on full precision.
results, metic
up
places,
namerically
D.10 CONVERGENCE ANALYSIS
Additional experimental results on the convergence of the BW discrepancy are available in Fig. 8. The BW objective consistently exhibits a monotonic decrease throughout the training process and reaches platcau after several epochs, thereby empirically validating the convergence of its optimization. In addition, we examine the evolution of MAE and MSE on the validation set. significant positive correlation is observed between the dynamics of the BW loss and both forecasting metrics (MAE and MSE). implies that minimizing the BW discrepancy effectively improves these forecasting metrics.
D.11 AUTOREGRESSION-BSEEI FORECASTING PERFORMANCE
Additional experimental results under the autoregression-bett based forecasting are available in Table 16.
D.12
PRORABILISTIC FORECASTING PERFORMANCE
Additional experimental results under the probabilistic forecasting setting available in Table 17, where we select D3U (Li et al., 2025b), the state-of-the-ar probabilistic forecasting framework as the testbed.
D.13 MULTI-SCALE FORECASTING PERFORMANCE
Additional experimental results under the multi-scale forecasting setting are available in Table 18, where we select TimeMixer (Wang al., 2024) and SCINet (Liu et al., 2022) as the testbeds.
E STATEMENT ON THE USE OF LARGE LANGUAGE MODELS (LLMS)
In accordance with the conference guidelines, we disclose our use of Large Language Models (LLMs) in the preparation of this paper follows:
24

---

Published as a conference paper at ICLR 2026
Table 14: Varying a results where Fredformer acts the forecasting model.
ETTh2
ECL
Weather
MSF
MAF
Disc
MSF
MAF
Disc
MSF
MAE
Disc
0.377
0.402
0.293
0.191
0.284
0.143
0.261
0.282
0.152
0.001
0.371
0.397
0.287
0.175
0.268
0.132
0.255
0.278
0.148
0.002
0.372
0.398
0.289
0.175
0.267
0.131
0.256
0.278
0.149
0.005
0.372
0,398
0.288
0.182
0.275
0.137
0,256
0.279
0,149
0.37U
0.397
0.285
0.257
0.279
0.02
0.369
0.395
0.286
0.182
0.275
0.136
0,258
0.280
0,149
0.05
0.370
0,396
0.285
0.187
0.279
0.141
0,259
0.281
0,150
0.1
0.371
0.397
0.288
0.196
0.287
0.148
0.261
0.283
0,151
0.2
0.372
0.398
0.290
0.209
0.298
0.158
0,263
0.285
0,152
0.376
0,399
0.292
0.230
0.317
0.171
0.266
0.287
0.153
0.386
0.406
0.299
0.239
0,326
0.177
0,268
0.290
0.154
Node: Bold and inderiined denote the best and socond-hest results. When metric values ooincide up to three decimall places, Hold indicates the numericllly ouperioe result based on full precision.
Table 15: The performance comparison of DF and DistDF different forecast models.
Models
TONec
TONet
TimeBeidge
TimeBriddee
Prediormer
Frediocmer!
Traasformer
l'Iandsommer?
FieTS
PreTs'
Merrics
MSE
MAL!
MSE
MAU:
MSL
MAE:
MSB
MAU
MSE
MAL
MSE
MAU
MSB
MAU
MSU
MAU
MSE
MAL
MSE
MAE
0.392
0.991
0.372 0.90
0.373
0.95
(.372
0.992
1.37)
0.900
0.333
0.391
0.335
0.405
0.981
0.403
10390
0.409
0.999 0.409
192
0.424
0.422
0.428
0.426
0.424
0.929
0.457
0.425
0.428
0:425
0.440
0,477
11434
0.451
0.442
0.457 0.447
336
0.486
0.454
0.472
0.442
0.471
0.451
0.467
0.450
0.485
0.449
0.466
0.445
0.450
0.457
0,476
0.455
0.501
0.472
0.504
0.474
720
0.487
0.488
0:453
0.904
0.608
0.557
|Avg|
[0.409
0.431
0.442
0.440
0.404
0.436
00.47
0
0.430
0.429
0.445
0.444
|0.459
0.474|0.47)
0.474
0.467
10:293
0.943
0.289
0.539
[02294
0.344
(0.259
0.308
10.293
0.544
0:267
0.356
0,301
0,349
0.296
0.347
10.115
0.574
(0.11 0,569
192
0.364
0.390
0.388
0.394
0.369
0.372
0.391
0.358
0:351
0.383
0,397
0.397
0.466
0.467
0.438 0.433
336
0.413
0.424
0.410
0.424
0.421
0.429
0.415
0.426
0.420
0:43
0.408
0.421
0.425
0.432
0.421
D.434
0.522
0.902
0.521 0.905
720
0:49
0:455
0.416
|Av 0.375
0.400
0.371
0.399
0.377
0,405
0.337
0.398
00:377
0.400
0.367
0.393
0.386
0.407
0.405
|0.524
0.496
0.466 0.467
10.310
0.352
0.311
0.331
[0.33
0.561
[0.319
0.388
10335
0.361
10.366
0.357
0.338
0.372
0.334
0.372
1o.342
0.375
1.335 0.371
192
0.356
0.377
0.377
0.585
0.363
0.365
0.352
0:358
0.380
0.382
0.396
0.397
0.355
0.400
0.379 0.303
336
0:988
0:400
0:987
0.400
n
0.908
0.408
( 999
0.405
0.996
0.404
0.990
0:404
0.437
0.424
0.415
0:188
0-416
0.421
n.408 0.415
720
0.,05
0.942
0.459
0.444
0:448
0:457
0.456
0,463
0.333
0.089
|Avg]
0.901
00:955
00336
0.400
[0:333
0.997
10.987
0.900
0.998
0.994
00.41
0.414
0.400
0.410
|0.1
0.421
|a 10.00 0.409
10.75
0.256
0.171
0.254
[0.177
0.259
[0.176
0.256
[0.177
0.260
0.174
0.256
0.182
0.265
0.181
0.263
10.188
0.279
la.155 0.275
1m2
0:343
0.342
0.298
0.257
0.264
336
0.209
0.336
01292
0.335
0.300
0.545
0.302
0.310
0.302
0.540
0.300
0.388
0.500
0,354
0.311
0.547
0.322
0.369
0,30 0.386
0.390
0.599
0.403
0.397
0.399
0.397
0.397
0.394
0.423
0.411
0.404
0.459
0.482
0.449
Avg
0:331
[02212
0.518
[02281
0.526
[0.280
0.325
10:000
0.534
0.207
0:321
0.396
0.0099
0.330
|0.366
0.965
Ja 1..00 0.368
10:14
0.33
00130
0.233
[.142
0.299
[0117
0.235
[0.161
0.258
10.145
0.258
1.150
0.242
0..48
0,239
[0.150
0.266
19.99 0.266
192
0.102
336
0.178
0.270
0.174
0.267
0.182
0,278
0.178
0.272
0.194
0.290
0.176
0.270
0.182
0.274
0.176
0.270
0.199
0,290
0.199 0,288
0.218
0.298
0.217
0.309
0.212
0.302
0.235
0.319
0.211
0.300
0.214
0.304
0.298
0.234
0.323
0.335
0.322
|Ag (0.175
0.205
00177
0.202
[01176
0.271
[0.122
0.267
[0.191
0.244
0.173
0.266
1.119
0.270
0.174
0.206
|0.199
0,208
(0.199 0.287
96
10:00
0.205
00:00
0.202
[1108
0.211
[0166
0.200
10.100
0.200
0.192
0.212
1.111
0.210
0.174
02214
10.19
0.206
(a 191 0.229
Weather
0.212
0.222
0.28
0218
0:255
0.346
0.256
0,266
0212
336
0.267
0.259
0.264
0.287
0.273
0.297
0.270
0.295
0.283
0.301
0.277
0.297
0.256
0,313
0.280
0.299
0.200
0.316
0.663 0,305
0.346
0.344
0.342
0.353
0.347
0.348
0.345
0.358
0.345
0.352
0:347
0.362
0.353
0.350
0.337
0.362
0.331 0.355
|Avg]] (0.246 0.270 [0.244 0,209 (a225 0.277
[0.248 0.275 J0.261 0.252 0.255 0.277
J0.209 0.289
0.258
0.280
|0.349 0.293
0.203|0.245 0,288
Not:: longth history window 96 all beselines. Aug indicates the resultts averagod over foroensinn langths: T-96, 192.
336 and 720. 1 marks the forecasting model transed DistDy:
25

---

Published as a conference paper at ICLR 2026
0,400
0.182
0.224
0.164
810
0.384
0.176
0.218
0.168
0.200
0.368
0.160
0.200
Epoch
Epoch
Epoch
(a) Disc on the training set.
0.345
0.925
BW
0.385
BW
0.135
0.330
0.325
0.875
Epoch
Epoch
Epoch
(b) Disc on the validation set.
0.552
0.792
0.260
0.783
MAE
0.549
MAE
0.774
0.346
0.765
0.250
0.543
0.758
Epoch
Epoch
Epoch
(c) MAE on the validation set.
1.325
0.170
MSF
0.878
MSE
MSE
0.672
1.255
0.160
Epach
Epoch
Epoch
(d) MSE the validation set.
Figure 8: Evolution of training objectives and validation metrics across four datasets: ETTm1, ETTh1, and ECL (from left right).
Table 16: The performance comparison of DF and DistDF on the autoregressive forecasting setting.
Models
TimeBridge
TimeBridge
Fredformer
Fredformer'
Metrics
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
0.405
0.402
0.395
0.391
0.391
0.396
0.386
0.390
E11ml
192
0.467
0.438
0.419
0.408
0.494
0.449
0.493
0,446
336
0.518
0.467
0.460
0.437
0.572
0.500
0.579
0.486
0.725
Avg
0.538
0.450
0.428
0.573
0.471
0.527
0.343
0.241
0.275
0.241
0.267
0.211
0.245
Weather
0.303
0.320
0.274
0.292
336
4,826
0.749
0.371
0.365
0.330
0.351
0.312
0.322
9.363
Avg
3.970
0.344
0.345
0.330
0.301
0.310
The length history window
buselines. Av' mirks the Foneenssing model trained via DiaDF.
averaged forecasting lengths:
T=96, 192, 336 and 720.
26

---

Published as a conference paper at ICLR 2026
Table 17: The performance comparison of DF and DistDF on the probabilistic forecasting task.
Models
DaD!
Metrics
MAE
CRPS
CRPS
MAE
CRPSam
0.357
0.263
0.723
0.357
0.720
ETTml
192
0.361
0,383
0.285
0.749
0.360
0.38.
0.282
0.747
720
0,460
0.437
0,325
0.892
0.453
0,435
0.328
0.849
0,383
0.395
0,293
0.776
0.380
0.394
0.293
0.762
0.176
0.240
0.174
0.179
0.173
0.225
0.171
0.173
192
0.223
0.271
0.205
0.234
0.217
0.265
0.198
0,210
0,309
0.253
0.269
0.278
0.310
0,260
720
0,359
0.361
0.273
0.419
0.353
0.360
0.269
0.378
0,259
0.295
0.221
0.275
0.255
0.290
0.218
0.255
Note: The length of history window 96 all boselines, Avrg indicatos the results averaged over foneessting lengthh T=96, 192, 336 and 730
marks the forecasting model trinsed via DisDR
Table 18: The performance comparison of DF and DistDF on the multi-scale architectures.
Models
TimeMixer
TimeMixer
SCINet
SCINet
Metrics
MSE
MSE
MAE
MAE
96
0.329
0.369
0.326
0.369
0.325
0.365
0.319
0.359
192
0.371
0.391
0.373
0.392
0.383
0.397
0.367
0,385
720
0.564
0.506
0.491
0.459
0.528
0.476
0.469
0.444
Avg
0.422
0.423
0.401
0.411
0.418
0.416
0.389
0.399
96
0.419
0.426
0.400
0.410
0.409
0.415
0.397
0.405
B37Th
192
0.464
0.451
0.439
0.436
0.457
0.441
0.448
0.434
0.455
720
0.614
0.553
0.501
0.486
0.505
0.482
0,501
0.479
Avg
0.501
0.476
0.456
0.446
0.467
0,450
0.459
0.443
96
0.159
0.145
0.242
0.248
0,242
192
0.161
0.258
0.159
0.256
0.167
0.266
0.159
0.257
BCL
0.277
720
0.212
0.302
0.207
0.298
0.202
0.298
0.197
0.294
Avg
0.176
0.273
0.172
0.267
0.173
0.273
0.169
0.268
96
0.173
0.220
0.168
0.217
0.160
0.208
0.158
0.207
Weather
192
0.213
0.254
0.212
0.253
0.214
0.257
0,21
0.254
336
0.286
0.306
0.273
0.298
0.276
0.300
0.271
0.298
720
0.377
0.362
0.354
0.352
0.362
0.356
0.359
0.351
Avg
0.262
0.285
0.252
0.280
0.253
0.280
0.250
0.278
Note: The lengih history window is set for basclines. indicaates the results averaged foneessting lengths: T=96, 192, 336 and 720. marks the forocasting model trained via DisiE:
27

---

Published as a conference paper at ICLR 2026
We used LLMs (specifically, OpenAl GPT-4.1, GPT-5 and Google Gemini 2.5) solely for checking grammar errors and improving the readability of the manuscript. The LLMs were not involved in research ideation, the development of research contributions. experiment design, data analysis, or interpretation of results. All substantive content and scientific claims were created entirely by the authors. The authors have reviewed all LLM-assisted text to ensure accuracy and originality, and take full responsibility for the contents of the paper. The LLMs are not listed as an author.
28


Reviews:
[{"review_id": "Njf2thwuIT", "reviewer": "Reviewer_ZpGW", "summary": "The paper addresses time series forecasting and proposes aligning the predictive conditional distribution with the true conditional distribution by minimizing the joint-distribution Wasserstein discrepancy. This approach mitigates the bias introduced by autocorrelation when using maximum log-likelihood objectives to train forecasting models.", "questions": "Please address my first point in the weaknesses. If no distributional discrepancy needs to be shown in the experiments, please elaborate why.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "* Good observation that both frequency and PCA components exhibit autocorrelation, which affects their learning bias; this provides a well-motivated basis for applying optimal transport theory.\n\n* The incorporation of DistDF in existing frameworks is straightforward\n\n* Comprehensive experiments; I appreciate the effort to compare with other distributional discrepancies and the application of DistDF to other approaches", "weaknesses": "* The central hypothesis of this work is that aligning conditional distributions is beneficial, and the authors provide theoretical justifications along with empirical evidence through forecasting error metrics. However, it is unclear whether the conditional distributions actually align for the best alpha values reported in Tables 5 and 6. In other words, the hypothesis is not directly evaluated in the experiments through distributional discrepancy, but rather indirectly through forecasting performance.\n\n* Improvements wrt to existing SOTA methods look rather small; however, they are consistent across models and datasets", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "mXMpxnggqC", "reviewer": "Reviewer_aY9s", "summary": "The paper proposes a Wasserstein-based discrepancy measure for time series that captures label autocorrelation and demonstrates the benefits of using it for time series alignment compared to established methods. Several experiments were conducted to support this claim.", "questions": "Perhaps I am missing something, but Table 1 seems confusing. DistDF, which is a discrepancy measure, is compared to other models. It should be pointed out which model was used with DistDF loss.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The presented Wasserstein discrepancy seems original and effective. The experiments seem comprehensive and well carried out.", "weaknesses": "I think the paper should discuss the assumption of Gaussian distributed data more. It seems absolutely necessary to derive the discrepancy measure and yet I suppose the benchmark datasets do not satisfy this property.\n\nI consider this a mild weakness but the theory regarding the general Wasserstein metric is presented mostly for discrete measures. Given that a Gaussian data distribution is assumed, it could be discussed how the presented results for empirical measures relate to the original Gaussian data distribution.\n\nMinor\n-------\nThe Bures-Wasserstein discrepancy is spelled as “Bruce-Wasserstein” in Lemma 3.5. Also in this Lemma, the equality to the W_2 metric should be made clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "oln3e5zXMt", "reviewer": "Reviewer_c2pc", "summary": "This paper proposes DistDF, a new training objective for time-series forecasting that aims to align the conditional distributions of forecasts and labels, rather than relying on point-wise MSE. Since conditional discrepancies are difficult to estimate from limited data, the authors introduce a joint-distribution Wasserstein discrepancy, optimized between the distributions of (history, labels) and (history, predictions). The method is model-agnostic and can be plugged into existing forecasting models. Experiments show performance improvements on multiple benchmarks.", "questions": "- Since the proposed objective must be combined with MSE for stable training, can the authors provide evidence that the improvement does not simply arise from a regularization effect? For example, how does the discrepancy term alone behave, and how strongly does its reduction correlate with forecasting accuracy?\n- The distinction between DistDF and existing learning-objective methods such as Time-o1, FreDF, Koopman-based losses, and Soft-DTW remains somewhat unclear. Can the authors more explicitly highlight the conceptual and practical differences, particularly regarding theoretical guarantees and optimization behavior?\n- The discussion of likelihood bias focuses primarily on MSE. Do similar issues arise in probabilistic forecasting frameworks using alternative objectives (e.g., quantile loss, CRPS)? If so, is DistDF compatible with or beneficial under such setups?\n- How well does DistDF extend to multivariate forecasting, probabilistic output formulations, or multi-scale architectures? Providing results or analysis in these more general settings would help verify that the proposed approach is broadly applicable beyond the current scope.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Strong and clearly articulated motivation regarding autocorrelation bias in likelihood-based objectives.\n- Solid theoretical foundation, including alignment guarantees and non-negativity properties of the objective.\n- Method is architecture-agnostic, enabling integration with a broad range of forecasting models.\n- Extensive benchmarking shows consistent improvements, supported by ablation studies demonstrating contribution of components.\n- Generally clear writing and clean presentation of the mathematical formulation.", "weaknesses": "- A key limitation is that the proposed discrepancy objective lacks guaranteed convergence or clear interpretability during training, making its practical effect on conditional alignment somewhat uncertain. Because the loss must be combined with MSE, the discrepancy may act more like a regularizer than a principled stand-alone objective. Additional empirical analysis of its optimization dynamics and correlation with performance would strengthen the claims.\n- More comprehensive experiments are needed to isolate the contribution of the proposed objective. Given that the method relies on a weighted combination with MSE, it should be compared not only against plain MSE training but also against other established time-series learning objectives (e.g., Dilate, Soft-DTW) when similarly combined with MSE. Such comparisons would help determine whether the observed gains stem from the specific discrepancy formulation or simply from augmenting the loss with an auxiliary term.\n- Evaluation is restricted to direct forecasting, limiting evidence of robustness across different training paradigms. Additional experiments under an autoregressive setting would be valuable to validate whether the proposed objective is broadly applicable across different forecasting architectures and training pipelines.\n- In Table 1, it is unclear which underlying model architectures DistDF is applied to. Since DistDF is a learning objective rather than a new architecture, and the table compares against architectural baselines, the presentation may confuse readers regarding what is being evaluated. Clarifying the base model used for each dataset would improve readability. Explicitly specifying the base architecture for each dataset (e.g., as done in Scaleformer, ICLR 2023) would improve clarity and ensure a fair interpretation of the reported gains.\nref. Scaleformer: Iterative Multi-scale Refining Transformers for Time Series Forecasting, ICLR 2023", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "hyiIQ6BzMR", "reviewer": "Reviewer_6qSb", "summary": "This paper proposes Distribution-aware Direct Forecast (DistDF), which achieves alignment by minimizing joint-distribution Wasserstein discrepancy between conditional forecast and label distributions to enhance forecast accuracy.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. This paper is well written and polished. Notations and equations are clearly presented and explained.\n2. This paper is well-motivated and offers an extremely thorough explanation.\n3. Experiments are comprehensive.", "weaknesses": "1. Experimental comparison (Tab. 2) lacks some most recent works, e.g., [*1]. The proposed method might not outperform these new works. TQNet [*1] achieves **0.377** MSE, **0.393** MAE on ETTm1.\n2. Experimental results could not fully support the significance of the method. The improvement is marginal when compared to prior art, e.g., TimeBridge, Time-o1, and TQNet [*1]. \n3. Improvement on presentation:\n - For results in the table, should not use **Bold** and $\\underline{\\text{Underline}}$ when two numbers are the same, use Bold for both.\n - (minor) In Section 4.3, the reference to Table 4 should be changed to Table 2.\n - (minor) Use consistent table style. Use \\toprule for Tab. 5 & 6\n\n\n[*1] Lin, Shengsheng, et al. \"Temporal Query Network for Efficient Multivariate Time Series Forecasting.\" Forty-second International Conference on Machine Learning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)