TEDM: Time Series Forecasting with Elucidated Diffusion Models
Authors: Edgardo Solano Carrillo
Sreerag V Naveenachandran
Julia Niebling
Venue: ICLR 2026 Poster (2026)
Keywords: Score-based generative models
Diffusion models
Stochastic Differential Equations
Time-series forecasting
Primary area: generative models

Abstract: Score-based generative modeling through differential equations has driven breakthroughs in high-fidelity image synthesis, offering modular model design and efficient sampling. However, this success has not been widely translated to timeseries forecasting yet. This gap stems from the sequential nature of time series, in contrast to the unordered structure of images. Here, we extend the theoretical formulation used for images to explicitly address sequential structures. We propose a diffusion-based forecasting framework (TEDM) that adapts score estimation to temporal settings and elucidates its design space. Such a design allows empirical computation of noise and signal scaling directly from data, avoiding external schedules. Notably, this reduces sampling complexity to linear in the forecast horizon. Without elaborate preprocessing, TEDM sets new state-of-the-art results on multiple forecasting benchmarks. These results illustrate the growing potential of diffusion models beyond vision. TEDM generates low-latency forecasts using a lightweight architecture, making it ideal for real-time deployment.

Published as a conference paper at ICLR 2026
TEDM: TIME SERIES FORECASTING WITH ELUCI- DATED DIFFUSION MODELS
Edgardo Solano-Carrillo, Sreerag V Naveenachandran, Julia Niebling Institute of Data Science, German Aerospace Center (DLR) (edgardo. sola aaaaanrrilll sreerag.. naveenachandran,
ABSTRACT
Score-based generative modeling through differential equations has driven break- throughs in high-fidelity image synthesis, offering modular model design and efficient sampling. However, this success has not been widely translated to time- series forecasting yet. This gap stems from the sequential nature of time series, in contrast to the unordered structure of images. Here, we extend the theoretical formulation used for images to explicitly address sequential structures. We propose diffusion-basee based forecasting framework (TEDM) that adapts score estimation to temporal settings and elucidates its design space. Such a design allows empiri- cal computation of noise and signal scaling directly from data, avoiding external schedules. Notably, this reduces sampling complexity to linear in the forecast horizon. Without elaborate preprocessing, TEDM sets new state-of-the-arr results on multiple forecasting benchmarks. These results illustrate the growing potential of diffusion models beyond vision. TEDM generates low-latency forecasts using a lightweight architecture, making it ideal for real-time deployment.
INTRODUCTION
Multivariate time-series forecasting drives critical decision decision-making across domains as varied as demand planning (Kamarthi al. (2024)), financial risk assessment, weather prediction (Oskarsson et al. (2024), stock market analysis (Zou et al. (2023)), and energy load forecasting (Symeonidis Nikolaidis (2025)). Unlike classical regression tasks, time-series data exhibit unique characteris- tics fies-trend, seasonality, and autocorrelation that demand models capable of capturing temporal dependencies and quantifying predictive uncertainty, particularly in high-stakes settings such as meteorology and finance (Box al. (2015)).
Recent advances in deep learning have dramatically improved forecasting accuracy by leveraging se- quence models. Transformer-base and Autoformer (Wu et al. (2021))-hane have consistently topped benchmark leaderboards. However, these approaches exhibit high computational complexity (quadratic time and memory requirements) (Kim et al. (2024); Kong et al. (2025)), and have poor long-term forecasting performance.
architectures in particular-aa suchh Informer (Zhou al. (2021))
Diffusion models have emerged as powerful generative paradigm across modalities, achieving state-of-the-art results in image, speech, and video synthesis (Xing et al. (2024); Ahsan et al. (2025)). Early attempts adapt diffusion modeling to time series, like TimeGrad (Rasul et al. (2021)), showed promise in computational complexity, by processing sequences using recurrent networks instead of transformers. Nevertheless, they fall short in forecasting long horizons. Other approaches followed (Yang et al. (2024); Su et al. (2025)), demonstrating that longer horizons are possible at the expense of more preprocessing and model complexity, additional the inherent sampling inefficiency of diffusion models (Ma et al. (2025)).
To strike a balance between long-term forecasting performance and computational complexity, a deeper understanding of the design space of diffusion models is needed. In the vision domain, this was elucidated by the EDM framework (Karras et al. (2022)) and imported to time series forecasting in climate applications (Price et al. (2025)). This allows optimization of noise and scale schedules, as well as time- dine-discretization strategies and solvers for the diffusion process. However, the sampling inefficiency remains, scaling O(SH) for S diffusion steps and H forecasting steps.

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Given
Forecast
Signal
ODE
What phttotte
Structured
Denoiser
step
Corrupted signal
Euler step
Euler step
Score function
Neural network
Denoiser
Diffusion across physical time
Denoised signal
Lunk
SDE
(a)
(b)
Figure (a) Denoising signals corrupted with structured noise. (b) TEDM uses score-based
generative modeling to forecast time series by numerical integration of ordinary (ODE) or stochastic (SDE) differential equation. The main characteristic of TEDM reducing the physical time and the diffusion time to the same axis, support by theory.
To address these limitations, we propose TEDM (Time Series Forecasting with Elucidated Diffusion Models), an autoregressive diffusion framework tailored for multivariate probabilistic forecasting. We extend the theoretical background of EDM for time series forecasting. Optimization the design space thus leads to significant reduction in complexity and increase in accuracy by:
Treating the diffusion and physical time axes the same (see Fig. lb). This reduces sampling complexity from O(SH) to O(H).
Using, for the first time, noise and scale schedules estimated empirically from the data. This avoids inductive biases from guiding the diffusion with artificially imposed schedules.
This allows TEDM to achieve state-of-the-arr results on several long-sequence forecasting benchmarks at a fraction the cost of traditional methods. We find that the best case for space complexity is OTT d) for given sequence of T timesteps and d features-ste giving promising results and then making our approach suitable for online settings.
RELATED WORK
Sequence Models. RNNs (Hewamalage al. (2021)) and TCNs (Chen al. (2020)) have been applied to capture nonlinear, temporal dependencies. Recently, Transformer- based vari- ants-Informme (Zhou al. (2021)), Autoformer (Wu et al. (2021)), DLinear, and iTransformer (Liu et al. (2024))aakkee long-range interactions via attention mechanisms and are dominating leader- boards. However, attention on sparse iregularly-sampled timestamps can degrade, and preserving temporal ordering remains challenging (Wu et al. (2021)). Additionally, most Transformer basoomeer-bbbedd forecasters produce only point estimates, limiting uncertainty quantification.
Diffusion models. Popularized by Score Matching with Langevin Dynamics (SMLD) (Song & Ermon (2020) and Denoising Diffusion Probabilistic Models (DDPM) (Ho et al. (2020)), they are promising for probabilistic time series forecasing due to their generative nature. TimeGrad (Rasul et al. (2021)) pioneered autoregressive score-based forecasting by combining RNN encoders with per-step diffusion sampling, though it inherits RNNs inefficiencies over long horizons. Non-autoregressive variants Seriants-such-ae as TimeDiff (Shen & Kwok (2023)) and TSDiff (Kollovieh et al. (2023)) generate the prediction horizon in step step-bypassing bypassing error accumulation and enabling parallel forecasting. ARMD (Gao et al. (2025)) improves autoregression and sampling complexity by supervising a devolution network that effectively learns to "jump" the S diffusion steps.
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Despite substantial advances, most of the existing diffusion-basee forecasters merely adaptations of image-domain DDPMs that do not completely harness the multivariate and temporal structures of time-series data. This is due to incomplete knowledge of the full design space of diffusion models that EDM helps to elucidate. In the following, we describe this background and introduce the changes needed for time series forecasting ultimately leading to TEDM.
BACKGROUND
3.1
EDM: UNIFTED DESIGN SPACE FOR DIFFUSION MODELS
Karras et al. (2022) present a unified framework for analyzing and improving diffusion -based generative models for image synthesis. Their contribution is the disentanglement of architectural, training, and sampling components into a modular design space, enabling independent optimization of each element. This allowed them to find optimal choices for each component and push the state of the art for image synthesis.
Deterministic sampling. In their formulation, sampling is grounded the probability flow ordinary differential equation (ODE)
da St at ot dr V log dt St
Here, 2t E Rd is the sample, o, is time-dependent noise schedule, 8t is a time-dependent scale schedule, and pe(a))) = s,=" an// / 0z) is the marginal distribution of the diffusion process. The latter is expressed in terms of a mollified version of the data distribution obtained by adding i.i.d Gaussian noise-of of standard deviation o- the samples. The term V log (x;) called the score function (Song et al. (2021)).
Deterministic sampling is achieved by integrating the ODE backwards, from time T where æy is completely noisy (øT maximum), to time 0 where is the prediction (og ~ 0). A related stochastic differential equation (SDE) adds noise during sampling for improved robustness. The authors Karras et al. (2022) propose using a second-order Heun's 's method with a linear schedule scaling 8; = which leads to smoother sampling trajectories.
f and constant
Denoising score matching. Training is based on denoising score matching: given clean data y ~ Pdata and Gaussian noise E ~ (O, I), the data is corrupted with unstructured noise n O,E. The training objective then minimizes the expected value:
E Ey-.i..e~.V(o.I) [I) n2; yl
This loss encourages the denoiser D(x; ox) to estimate the conditional expectation of the clean signal given the noisy input y as well as the noise level de is related to the score function (see Eq. (3) of Karras et al. (2022)) by:
Va 1og D( -
of
Preconditioning. To improve stability and expressiveness, the authors propose a preconditioned architecture for the denoiser Dg:
Do(2; = (a) x (a) Fo (Cuu(()) Cnoise (o)).
Here, Fg is the core neural network to be trained, and Cakip; Gin, Couty Cnoise scalar functions of o that control signal scaling and conditioning. These are derived analytically by requiring the network inputs and training targets to have unit variance (Cis Cour), and amplifying errors in Fø little as possible (Csss)). Except for Cnoise (which is chosen empirically), these functions are expressed in terms of - = Var(y), which is uniformly set in image datasets. The authors also propose log-normal distribution for sampling noise levels o during training, along with loss weighting X(o) = 1//2m((0) This modular reformulation facilitates targeted improvements and enhances compatibility with a range of generative architectures.
Importantly, this unified framework subsumes earlier methods like DDPM. DDIM, and SMLD (Song al. (2021)). These differ mainly in their a, and 8; schedules, time discretizations, and
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preconditioning schemes. For instance, DDPM uses di t, 8t (as EDM), and stochastic sampling. DDIM replaces the stochastic reverse process with Euler integration the same ODE and time steps DDPM. SMLD instead models the score function directly with variance-exploding SDE. This modular view reveals that improvements in training or sampling can often be transferred across models without retraining the network.
3.2 EXTENDING EDM TO MULTIVARIATE SERIES
Adapting EDM to multivariate time-series introduces unique challenges present in image domains. First, applying a shared noise schedule across all features assumes uniform scale and dynamics. In practice, features may differ in variance or predictive importance, making uniform noise injection suboptimal. Feature-specific noise scaling is needed. Second, time-series exhibit strong temporal dependencies. EDM's i.i.d. Gaussian noise assumption can disrupt autocorrelated patterns, especially when noise is added independently at each time step. Structured noise or causal conditioning mechanisms may be required to preserve temporal coherence. Finally, architectural design must respect the sequential nature of time. Preconditioning schemes should account for temporal scale and position. Temporal encodings, autoregressive models, or attention basod architectures may be better suited than standard convolutional backbones. We discuss next how these challenges are thcoretically and experimentally addressed by TEDM.
4
TEDM METHODOLOGY
4.1 PROBLEM DEFINITION
Given multivariate series Y1:1 RCXT, with C' features and T time steps, the problem forecast the next H steps. The forecast is done through a mapping fe YI:r + YY++IT+H- consisting of learned estimator the score function and its autoregressive use in an ODE (or SDE) solver. Unless otherwise necessary, we omit the subscript from the respective variables. The theoretical results in the following are derived in appendix A.
4.2 DATA-DRIVER NOISE AND SCALE SCHEDULEES
We extend the EDM formulation to multivariate noise schedules. With E, = 8, Cov( the forward ODE in (1) takes now the general form
dx: Št 2. 1 E, V lo.))(x.).
(4)
dt
St
We restrict to deterministic sampling in the following and leave probabilistic forecasts (based on our SDE (A.65)) for Appendix D. In formulation, the score function becomes (Eq. (A.25))
Va (x.,) s; (p<r//s;; E,) - xz/ss].
(5)
and allows the backward ODE associated to Eq. (4) to be written as (Eq. (A.54))
dx, - log s,) x; (dE, Σ,1 D(x,/s, Σ,) x2 st].
(6)
This expression suggests our proposed contributions:
Since dt does not appear in the difference equation, we do not need any strategy to quantify time increments, needed by all previous approaches. As consequence, we take the physical time-axis of the time series as the time-axis of the diffusion process.
Diffusing across the time-series horizon implies that the noise E, and scale 8, schedules acquire physical meanings. Unlike any other diffusion model far, we empirically estimate these from the data.
The way to estimate these schedules is suggested from their definition. We can show (Eq. (A.5)) that the scale 8, obeys [[(x)) 8t E(xr)) Furthermore, show (Eq. (A.7) that Cov(æ,) of Σ.
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Therefore, estimate s, and Σ, from the input data window y1.7, by empirical estimations of the above relations. We follow two approaches:
Cumulative estimation. We estimate s and E, from the cumulative mean and covariance of the data, respectively. That is, B(x,) Mean(y)ac) (x)))
yiul:
- Mean(y):)) Viai
(7)
where the division is element-wise and the reduction is along the sequence axis. Similarly, we estimate E, from the cumulative covariance the data,
Cov( Cov(y1:),
Σ. = Cov(y1a) ST
(8)
where S; ( $T,)) a matrix, build from (7), that applies congruent scaling to the covariance matrix, preserving its positive definiteness.
Sliding window estimation. We estimate st and E, from the mean and covariance of sliding window over the imput y1:T. This allows more flexibility to adapt to local changes in the data statistics. It also avoids the problem of defining the variance for the first data point of a window sandow-technicalyy zero, so interpolated in the cumulative estimation above. Additionally, helps mitigate issues with data values (Y1.1 in Eq. (7)) close to zero that may blow up the cumulative scale estimate.
4.3 TRAINING
As with EDM, we train using denoising score matching. Given subsequences clean data y ~ Pdata- compute the associated subsequences of empirical E. Gaussian noise (N(O, is drawn and structured with this schedule: subsequence is corrupted with a different noise level, the noise is no longer i.i.d. as in EDM. The denoiser learns to remove this noise (see Fig. 1a), by minimizing
/22 Since every time step (and feature) in the data
E P- DI + n; E) yl
We evaluate different architectures for the denoiser: They differ in the way the temporal structure leveraged and whether conditioning past data is done. We extend the preconditioning scheme of EDM to matrix- E. For this, the denoiser is expressed as (Eq. (A.27))
Do(æ, E) Syiskip Cx Fo (Zzzi æ; Cx ).
By imposing that the inputs and training targets of Fo have unit variance, and that its errors are amplified as little as possible, we have (Eqs. (A.35)):
C = (Cov(y) + Σ) 11/2
Cx = (y) (B))(()Cv (Cov (y) + E) -1
L = Cov (oo(y)(ovv) + -E)- -1-EE
x 1/c2
The matrix Canose is chosen empirically, as in EDM, and A weighs the loss function in Eq. (9). Note that these expressions reduce to those of EDM when == = Furthermore, they hold even when Σ is estimated from y. provided that the estimator is unbiased.
4.4 INFERENCE
Once the denoiser is trained, the score function is estimated by (5). Knowing the score function provides the mechanism for sampling by ODE (or SDE) integration. Since the diffusion and physical time axes are the same, this allows forecasting the next time step by an Fuler step of (6) (Eq. (A.55)):
ザ = I log
St St_1
9. + 1 (E,- - E, E, (D+(ū./s Sss.. E,) - 0t//sll
where (.)+ denotes the projection onto the cone of positive replaced by their estimates in (7) and (8) (or their sliding window counterparts). This inference
matrices. Here, s, and Σ,
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Budh
Eute
Blui
Elum2
0.9
0.3
-.5
3.0
-2.0
Excisange
Sotar
Steck
Wouther
1.6
1.0
0.8
0.0
0.2
20
20
610
60
20
92
Figure 2: Random test windows from all datasets. Each subplot corresponds to one dataset and shows 5 windows from randomly selected time indices and feature dimensions.
rule is general, and can be implemented by a Cholesky factorization of Σ, that allows applying E, to vectors. The projection to the positive semi-definite cone requires diagonalization of Σ; Et-1. which may be costly for large feature size d.
An efficient approximation is obtained when the noise covariance is considered diagonal, i.e. Σ, = diag( 1 .,,.......)). In this case, the inference rule simplifies considerably, since all matrix operations reduce to clement-wise operations the feature dimensions. The Eurler step in this case becomes (Eq. (A.53))
F++1 = log
St Σ 81-1
1/2 8
log 2, 1/2 De( E.).
(10)
Remarkably, this inference rule holds exactly beyond the diagonal approximation (see Remark A.1), as long as the principal axes of Σ, do not change with time (only its cigenvalues do). This applies to systems with stable directions of variability but evolving intensities along those directions e.g. spatial climate patterns, brain signals with stable spatial modes, etc. Due to its simplicity and efficiency, we consider the diagonal approximation for the rest of this work.
Starting from the given data window 71 := y1.7, an Euler step then transforms this window to predict D2 : 1/27+11 Assuming = H (the horizon length), after Euler steps (see Fig. lb), we get the predicted window D1 + I1+ Thus inference time is O(H) instead of O(HS) for S diffusion steps, since one Euler step replaces one diffusion step.
Intuitively, every data point in the given window (a single imagined to particle "propagated by diffusion" to a corresponding data point in the predicted window. All these particles in the window processed in parallel by the network that learned to denoise the signals.
y/ in description) may be
EXPERIMENTS
5.1 DATASETS
We evaluate our approach on eight widely used multivariate datasets: ETTh1, ETTh2, ETTml, ETTm2, Exchange, Solar, Stock, and Weather. The Traniity-Traanffmm---eeeeeee Temperature (ETT) datasets (Zhou et al. (2021)) are standard long-sequence forecasting benchmarks, sampled hourly (ETTh) and every 15 minutes (ETTm) from two power-transformer stations. The Ex- change dataset (Zhou et al. (2021)) contains daily exchange rates across major currency pairs.
Table 1: Comparison of benchmark time series datasets: data size and feature count.
Dataset
Timesteps
Dim (d)
ETTh1/ETT12
17,420
7
ETTm//ETTm2
69,680
Exchange
7,588
8
Weather
52,695
21
Solar
52,560
137
Stock
4,431
The Solar dataset (Lai et al. (2018)) consists of 10-minute solar power measurements from 137 sta-
tions, and the Stock dataset provides daily Google stock prices from 2004-2011 (Yoon al. (2019);

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Yuan & Qiao (2024)). Dataset sizes and feature counts reported in Table 1. Randomly selected test samples are shown in Fig. 2. Further dataset details are provided in Appendix C.I.
5.2 DATA PREPROCESSING
known that time series forecasting methods requiring extensive preprocessing typically un- derperform simpler methods (Makridakis et al. (2020)). Such preprocessing include stationarity transformations (Liu et al. (2022)), seasonal adjustments (Findley & Monsell (2019)), multi- multi-resolution analysis (Li et al. (2024a)), etc. These steps often make the approaches cumbersome and introduce inductive biases.
Our method does not require complicated preprocessing. The datasets are partitioned into train (70%), validation (10%), and test (20%) sets, according to the standard practice for these benchmarks (Liu et al. (2024)). Following Gao et al. (2025), we apply Z-score normalization to the data. We further partition the sets into subsequences yI.r (windows). For training, each window has = (horizon length) time steps, consumed by the denoiser network in batches. For validation and testing, padding is added for TEDM to compute time shifts as appearing in Eq. (10). Each window has 2H time steps (plus padding and additional context for conditional denoising). The first T time steps are given to the models for forecasting, the last H steps are the ground truths to contrast with the model predictions. We fix H 96 timesteps for all datasets, in Gao et al. (2025). See Appendix E for longer horizons.
5.3 BASELINES
We are inspired by ARMD (Gao et al. (2025)) and follow their experimental setup closely. However, we include newer diffusion baselines that were not available at the time of their publication. In particular, NsDiff Ye et al. (2025) is a recent approach that extends diffusion models to non-stationary time series, providing a strong baseline for our experiments. They provide a codebase evaluate TMDM Li et al. (2024b), DiffusionTS Yuan & Qiao (2024), TimeDiff Shen & Kwok (2023), among others. We use their codebase and experimental setup to evaluate these methods for H T 96.
5.4 METRICS
We follow the evaluation methodology of Zhou al. (2021) and use the mean square error (MSE) and mean absolute error (MAE) to measure performance when comparing z-score normalized data. This has become standard practice in time series forecasting (Gao et al. (2025)). For probabilistic forecasts in Appendix D, we use the standard continuous ranked probability score (CRPS) (Matheson & Winkler (1976)) and quantile interval coverage error (QICE) (Han et al. (2022)).
5.5 NETWORK ARCHITECTURES
One of the advantages of elucidating the design space of diffusion models is that the selection of model architectures is more intuitive, The denoiser (as an estimator of the score function used for inference) has job independent from the forecasting task (see Fig. Ia). This separation makes it ideal for generalization, and gives more flcxibility when designing architectures aimed at better denoising outcomes. We considered several architectures, ranging from the simplest Linear network, with OT d) space complexity, to the most complex UNet architecture. These architectures optionally accept condition past data, support conditional denoising (Batzolis et al. (2021)) )). For more details on these architectures, see Appendix C.3.
5.6
MAIN RESULTS
Table 2 shows that TEDM delivers state-of-the-ar accuracy across the majority of datasets, achieving the best MSE and MAE on ETTh2 (0.214/ / 0.319), ETTm2 13.13// / 0.253), and Exchange (0.069) 0.183). On ET TEDM ranks second (MSE 0.419, MAE 0.421) behind ARMD, and on Weather, TEDM is also second (MSE 0.223, MAE 0.261), close to the strongest baseline
that dataset.
The only dataset where TEDM trails the simpler TimeDiff/ARMD pairing is ETTh1 (MSE 0.595, MAE 0.524), which we attribute to typical large-and amplitude changes (see Fig. 2) that stress TEDM's assumption of smooth flows (see Assumption A.1). Code is available at https: / ////ttttt/t /gitlab. com/ dlr-dw/tede tedm.
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Table 2: MSE and MAE scores (prediction horizon H 96) for diffusion-based forecasting methods.
Best scores per dataset are in bold; second best are underlined. Lower is better.
Methods
Metric
ETTh1
ETTh2
ETTml
ETTm2
Exchange
Weather
MSE
0.417
0.364
0.548
0.209
0,208
0.228
TimeDiff
MAE
0.456
0.393
0.485
0.296
0.331
0.305
MSE
1.032
3.017
0.976
3.517
3,302
0.625
DiffusionTS
MAE
0.757
1.340
0.726
1,472
1,493
0.609
MSE
0.534
0.564
0.421
0.313
0.212
0.180
TMDM
MAE
0.514
0.517
0.408
0.350
0.338
0.241
MSE
0.445
0.311
0.337
0.181
0.093
0.232
ARMD
MAE
0.459
0.338
0.376
0.255
0.203
0.291
MSE
0.552
0,460
0.450
0.250
0.146
0.223
NsDiff
MAE
0.506
0.452
0.434
0.328
0.280
0.276
MSE
0.595
0.214
0.419
0.135
0.069
0.223
TEDM
MAE
0.524
0.319
0.421
0.253
0.183
0.261
TEDM
NsDIff
TMDM
DiffusionTs
TimeDifr
1.00
2000
1.00
200
100
1.00
200
contaxt
Figure 3: Models evaluated on sample from the ETTm2 test set.
Table 3: MSE and MAE scores (prediction horizon H 96) for non diffusion- forecasting methods. Best scores per dataset are in bold. Lower is better. Results taken from Gao et al. (2025).
Methods
Metric
ETTh1
ETTh2
ETTml ETTm2 Exchange
Solar Stock
MSE
0.386
0.297
0.334
0.180
0.086
0.203
0.342
iTransformer
MAE
0.405
0.349
0.368
0.264
0.206
0.413 0.413
MSE
0.384
0.340
0.338
0.187
0.107
0.427
0.427
TimesNet
MAE
0.402
0.347
0.375
0.267
0.234
0.499
0.499
MSE 0.386
0.333
0,345
0.193
0.088
0.286
0.286
DLinear
MAE
0.400
0.387
0.372
0.292
0.218
0.325
0.325
MSE 0.414
0.302
0.329
0.175
0.088
0.516
0.516
PatchTST
MAE
0.419
0.348
0.367
0.259
0.205
0.524
0.524
MSE 0.392
0.305
0.336
0.184
0.086
0.352
0.352
Client
MAE
0.409
0.353
0.369
0.267
0.206
0.433
0.433
MSE 0.595
0.214
0.419
0.135
0.069
1.061
0.056
TEDM
MAE
0.524
0.319
0.421
0.253
0.183
0.662
0.182
These quantitative gains mirrored by the qualitative behavior in Fig. (more in appendix G): TEDM tracks target trends more faithfully, with better phase alignment and amplitude calibration, and fewer spurious oscillations than competing diffusion models. We also compare with mate-of-thee. oftthe--rt non-diffusion methods (see Table 3), confirming the superiority of TEDM for several datasets.
These results indicate that elucidating the design space decoupling the denoiser from the sampler while carefully selecting the schedule and integration strategy-confer confers consistent advantages in performance across diverse temporal patterns.
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Table 4: Ablation study of elucidated models. Lower is better. Percentage gains (in parentheses) indicate improvement of TEDM over EDM.
Methods
Metric
ETTh2
ETTm2
Exchange
MSE
0.730
0.756
1.276
IDDPM+DDIM
MAE
0.657
0.664
0.963
MSE 0.419
0.293
0.448
EDM
MAE
0.495
0.405
0.532
MSE
0.303
(28%)
0.137 (53 %)
0.110 (75%)
TEDM (cumulative Σ,, 81 1)
MAE
0.377
(24%)
0.249 (39%) %)
0.241 (55 %)
MSE
0.242
(42%)
0.135 (54 %
0.068 (85 %)
TEDM (cumulative Σ, empirical st)
MAE
0.337 (32%)
0.250 (38 %
0.181 (66%)
MSE
0.216
(49%)
0.142 (52%)
0.075 (83%)
TEDM (sliding E, empirical s,)
MAE
0.259 (48%) (48%)
0.249 (39 %)
0.195 (63%)
1.0
0.2
0.0
0.0
0.9
0.2
2..0
EDM
IDDPM
Observed
TEDM
Figure 4: Forecasts generated by EDM, iDDPM+DDIM, and TEDM on four randomly selected subsequences from the ETTm2 dataset. Only a section the initial subsequences for the right panels observed for better comparison of the forecasts.
5.7 ABLATION STUDIES
We adapt the EDM work of Karras et al. (2022) from images to time series. The inherited modularity allows us try different noise schedules, time discretization and sampling strategists. For instance, EDM, -DDMM-DDIM, and TEDM, all in a unified codebase. We refer to these as "Elucidated" models, since they fit into general, modularized diffusion framework, mirroring the terminology from Karras et al. (2022).
Our extension of the EDM framework time series gives results consistent with those in the vision domain. That is, EDM is consistently better than iDDPM+DDIM. by harnessing optimization of the design space. For time series, this a result already leveraged by the best weather forecasting framework to date (Price et al. (2025)). Our contribution goes further in two main dimensions:
1. By considering empirical, rather than preset, noise (Ee) and scale (8t) schedules, we get performance gains (see Table 4) of up to 85% in MSE (66 % in MAE) with respect to EDM. Fig. 4 shows qualitative results.
2. By aligning the diffusion and physical time axes, significantly reduce the time complexity of sampling, getting resource benefits comparable to ARMD (see Table 5). The latter does not harness the optimization of the design space though, shown in the main results.
We also evaluated the loss of skill when considering different forecast horizons. This is shown in Table 6 and compared against the classical skill (Hyndman & Athanasopoulos (2018)) of a forecast that extrapolate the mean of the context window (see appendix B). This is the minimal-skill forecast (yet better than random). TEDM is still able to leverage pattern-wis information to forecast much better than the latest observed average. For comparison with other methods, appendix E.

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Published as a conference paper at ICLR 2026
Table 5: Average per-batch training and inference time (seconds), memory (MB), and test MSE
ETTm2. Lower is better.
Method
Train Time (s)
Train Mem (MB)
Test Time (s)
Test Mem (MB)
MSE
TimeDiff
0.022
759
21.38
125
0.209
DiffusionTS
0.098
3112
634.96
14595
3,517
TMDM
0.207
15600
26.83
193
0.313
NsDiff
0.107
2682
9.80
1125
0.250
ARMD
0.009
20.7
0.02
21.3
0.181
TEDM
0.004
21.3
0.11
23.9
0.135
Table MSE and MAE scores for TEDM on ETTh2, ETTm2, and Exchange for longer forecasting horizons, with baseline (mean) forecast errors. Lower is better.
Horizon Metric ETTh2
ETTm2
Exchange Baselinemean
MSE
0.216
0.132
0.068
1.010
MAE
0.321
0.251
0.182
0.801
MSE
0.260
0.163
0.153
1.005
192
MAE
0.354
0.282
0.276
0.800
MSE
0.326
0.248
0.283
1.003
336
MAE
0.396
0.351
0.382
0.799
MSE
0.528
0.298
0.602
1.001
720
MAE
0.510
0.386
0.571
0.798
LIMITATIONS OF THIS WORK
Although the theoretical foundations of TEDM have general scope within the diffusion framework, all time series can be represented as the Itô processes underlying such framework. For instance, our formulation cannot capture memmoyy dynamics such as those exhibited by fractional Brownian motion (Mandelbrot & Van Ness (1968)). Similarly, our framework cannot represent heavy-tailed or power-law noise (e.g.. a--tablle processes; Samorodnitsky & Taqqu (1994)), nor the jump-driven behaviors (Applebaum (2009)), all of which violate the diffusion regularity assumptions. Furthermore, its effectiveness was shown in the diagonal approximation of the data covariance, which most likely breaks down for datasets with high-dimensional feature space (e.g. Solar in Table 3).
7 CONCLUSION AND FUTURE WORK
We present TEDM, the first time series forecasting framework that fully elucidates its design space, grounded on solid theoretical background. This allows TEDM to reduce the computational complexity levels suitable for online deployment. We plan to extend our work with more detailed analysis of the skill in probabilistic forecasting, including a method to sample prediction intervals without ensembling. Aditionally, we foresee the usage of TEDM for anomaly detection. data compression and imputation tasks.
USE OF LARGE LANGUAGE MODELS
We used large language models (LLMs) only to polish the writing and to help search and organize related work. No modeling ideas, algorithmic designs, experiments, analyses, reported results were produced by LLMs; all technical content and empirical results were created and verified by the authors.
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APPENDICES
In the following, we provide theoretical derivations and experimental details of the main results in the paper.
A THEORETICAL FOUNDATIONS OF TEDM
A.1 PRELIMINARY
Following Song et al. (2021), we consider diffusion processes as solutions to an Itô SDE
dx, = f t) dt + G( t) dwi.
(A.1)
where f(:tt: : RR + Rd and G(.,t) RR Rdxd Here, w, a Wiener process, with changes having zero-mean, B(dw;) O, and being uncorrelated: E(dw,,ww?)) Idl.
To simplify notation, we label functions of time with a subscript, e.g. the drift term is written as ( () f(x)). Also, explicitly keep the time dependence of a,, when important from the context, otherwise we just use x.
We consider the diffusion term, G(æ,, t) G., be independent of (or slowly varying with æ)) See Assumption A.I for a more concrete statement about this condition. The SDE then becomes
dx; = f(()) + G. dur.
(A.2)
In the following, we consider the case of affine drift term, for which the perturbation kernel is Gaussian.
A.2 PERTURBATION KERNEL
The transition probability density for drift terms of the form f((x) f.æ, with f. : R + R, is Gaussian (Eq. 29 in Song et al. (2021)):
Por D (o) = (a Pts Vi),
(A.3)
with mean na and covariance V, To find these moments, we express (A.2) in the form
dx, = fiæ, dt + Ge dws.
(A.4)
Taking expectation value
dIE();) = f(E(x:) dt + G, E(dwi)
f(E(x:) dt
dpp Susdl
dup filt
dt
be So f-dr po S1D0
where the scale process is defined as Using this, the integral form
E(v,) 8t E(xo),
= edd fedt
(A.5)
2t = 8:00 + I' s- G.wrr
(A.6)
j0
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of the Itō process (A.2), find the covariance
Vi Cov(x;)
1.'00 8, G,dlur
Cov
2
Cov(G,dd.))
2
8T GG Cov(dwr)
it
3
ST G,
0
87 En
(A.7)
where we have defined
Σ. S 5t G.
(A.8)
o G.G..
(A.9)
The perturbation kernel (A.3) can then be written as
Por ( = N (ax;;; 3,00, si E.
(A.10)
A.3 SCORE FUNCTION AND DENOISER
The score function is defined V log Pr (æ). where pz( is the marginal distribution. The latter is obtained by integrating (A.10) over all initial conditions:
(x))
Poe (xc (pp) Paaua (xo)
gd
Nax: seao s Ee) Paaaa (Do) dxo
Jgd
Nwx/ (ei//81 wo: Ex) Paata (2rg dayo
p(a0/ 88, Et):
(A.11)
where we have used the fact that and therefore express
(2m) 1 1/2 ()--)) (a--1)
N(
(A.12)
N(x 8;00, sf Ex) = st (x///// st Zo. E.).
(A.13)
Following Eq. 19 in Karras et al. (2022), we have defined the mollified version of the data distribution
p(x, (x, Σ) = Pdata (x) (0; E).
(A.14)
as convolution that effectively corrupts data samples with Gaussian noise.
Score function. The score function is calculated from (A.11)
Va log px())
= Va log[oo p(r/s Σx)] V.
E,)
Vap(x/s /s, E.)
p(æ/ St, Ex)
s, Va p(ă, Ex)
à x 8t-
(A.15)
p(ă, Ex)
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To evaluate this, we need an analytical expression for the probability density of the data.
Consider a dataset with a finite number of samples {y,,... yn j. Assume that these arise from transforming the dataset {-1.-.. .NN using continuously differentiable mapping g. so Y g
Ei=1 Pæ, 6(x æ;), where Pæ, is the probability mass of The discrete dataset has density fx (a:) Σ N The density of Y is known to be
fr(y) | fx(x))6y g(xx)) dar
(A.16)
Pzx
8(x æ:))
ö(y - -0( dæ
(A.17)
1==1
N
Pa.
8(x x0)0(1 - g(x)) dx
(A.18)
Agd
1=1
Pm. 50y g(xx))
(A.19)
N
Pn. Pny5(y vi).
(A.20)
In the case of images, g is the identity mapping and Pæ; = 1 I/N is the uniform density. For time- series, think of a propagator that takes each æ; into the corresponding y; after number of physical time steps. One can then write the data distribution as
N
Pdutta (D))
Σ 2=
Pa. 5(z0 - y-)-
(A.21)
With this, we can evaluate (A.11) as
/
pl
Ex)
Nie; wo; Ee) Pautaa (ang)
Rd
N(z; wo. Ex)
Σ Pae 8(x0 - 4i) doc
Rd
N
Σ
Ps. N(;: yr, E.)
(A.22)
i=1
N
Va p(. Ex)
Σ
Pw. V Na;: yú E.)
(A.23)
i=1
From (A.12) we have
VaN(e; = -N(:; Ex log Ni Ex)
=N(d;y, E.) 1- >, (a -
=N(d;y E.) ( -
Substituting in (A.23)
N
Va p(2, Ex) =
Σ
Pa; Na: Σ.) --- ( - y:)
i=1
(A.2)
D (p Σ.) D è. Σ क p(C E.)
= Σ [D
$ Σ,) a
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where we have defined
Li=1 Pa, Pa, ((; yss E1
D@ē, Ex)
(A.24)
Ei=1 Pe, (2; yss Ex)
Substituting in (A.15)
V m logp:(z)) = Ex [D Ex) x/ss].
(A.25)
Denoiser. We want to show that (A.25) links the score function to the denoiser. That is, (A.24) is the optimal solution the denoising objective
((D; E) = Eyr-pan pam En- D + n Σ) y|l
Eum pum Eow D(x, )) y|
Pen Eyr-pen
V(x:y, Σ) ((x, Σ) 2
Bd
N
Na,
D(a Σ) y: di
nd
Jed L(D; a, Σ) da.
Since the integrand is positive everywhere, the optimal solution D, satisfies
(. Σ) arg min L(D; x. E). (
Since this is convex optimization problem, the unique solution is found as
0 V D( L(D; E)
Σ Ne, D(x, Σ) y:l12
Σ Pa, No; y. Σ) V D E) y.1l2
Na: yı. 2)2(D, E) y.]
i=1
Ei Pa. Na, Nax; ys E)
(a Σ) =
Ee Par ((x; ye, Σ)
This optimal solution agrees with (A.24).
A.4 TRAINING WITH PRECONDITIONING
Motivated by Karras et al. (2022), we train the denoiser (n E) by minimizing a weighted version of the denoising objective
((D,) Eg- X L(Dg; E),
Eg Ey En (n) Do + -
(A.26)
where Ps is distribution over noise covariances, and X is weighting function. For simplicity, we write single functions of Σ in terms of subscripts, e.g. X = Xx
We extend the EDM preconditioning strategy as
Do(ar, Σ) C. Cxvut Fo Criin 8; Cx.aisee ):
(A.27)
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where Fo (+ -) is neural network with parameters 0; Essipp Crinn and moiie preconditionigg matrices that depend on the noise covariance Σ, and Cxuuu is a scalar. Substituting (A.26)
((0) y. ne Az Czakiip y + n²) CE Fo (C + n)): Einoise) y11
Ex n Ax Saal Fi CE (y + n);; Exnoise))
2
CEioni (y Ch + n))ll
y vn d Fo C (y + rt); Cinoiis)) Larrget(33 (3, nR, Σ)
(A.28)
which is just the 12- 12-supervision of the neural network Fo(::) to match the target function. While doing this, it is required that:
Fa sees unit-variance inputs. This is achieved by choosing Cz such that
Cov(Cr (y n)) I, Cziin Cov + n) C I,
Cz (Cov(y) + Σ) C Ejin I, C (Cov(y) + (+))))=)-111 -/11//
(A.29)
Note that this holds even when estimated from the data y. as long as the estimator is unbiased. This is because, in general, from the law of total covariance,
Cov(y + n) E[Cov(y nl + Cov(E nly)) Cov(y) + E[E],
since Cov(y)3) 0, Cov(nly) Σ, and Ey + y)] y.
Furget gives unit-variance outputs. This is achieved by choosing Cx such that
Ciagee (,, Σ)) I, CE
Cov (y Cz (y + ni)) I,
Cov ((I Crnniipy - Criskipp) = I,
Exout
(I Cov(y) (I CZ )"2 Chiskip EC = aan
(A.30)
Errors in Fg are amplified as little possible. This is achieved by choosing Sakip above to minimize Cx For this, we can set up the optimization problem
min Czatip
Eautt
s.t.
(zaNp)) Cov(y) (I C2 )"
Crapki
EC CC
2 J.
2
This constrained matrix optimization problem that can be solved with Lagrange multi- pliers. That is, we define the Lagrangian
L(Csi Cxout A) Souu
+ Tr A ((I CxaLiP) Cov(y) (I Criskip)
+ Craskip C - 2 1)
where A is symmetric matrix of Lagrange multipliers. Setting the gradients to zero gives the optimality conditions
V C L = -2A (xiskip) Cov(y) + 2A Cε Σ 0,
V Vorion C = 20E 20 Tr(A) = 0,
VA L (I CΣ.aki) Cov(y) (I C. Craskip E CT Eiskip
S I (A.30)
0.
From the second condition, we have that either Cxsouut = 0 or Tr(A) = 1. The former is not acceptable, since it would lead trivial solution. The latter implies that A A 0. Therefore,
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the first condition can be written
-2(1 CIiskip) Cov(y) + 2 Si E = 0,
(I Esisii) ) Cov(y) Cs E,
(A.31)
Cs (Cov + Σ) = Cov(y);
Ciskipp = Cov(()(Cov(t)) + -)-
(A.32)
Substituting (A.31) back in the constraint equation (A.30), we find
2 (I (yskip)) Cov(y) (I Cxiskip) ) Crisiip E E
= Exout I
Sakiip E (I- Chisiip) Ejskip E ΣC E E I
2
Criskip = 2 Eour I
Cov + 2) -) (A.32)
Eou I.
(A.33)
With Ay Cov(y) - E, that Ag Sou so 2 is the eigenvalue of Ay with eigenvector e. To minimize Coouuts it must he chosen as the smallest eigenvalue of Ax.
The loss weighing is uniform. This is achieved by choosing ' such that
A cont 1,
(A.34)
Ay is the inverse of the smallest eigenvalue of Ag.
The preconditioning then ensures that the neural network F6 is trained unit-variance inputs and targets, while minimizing the amplification of errors and ensuring uniform loss weighting.
Preconditioning summary. The preconditioning matrices and scalars are
Cy (Cov(y) + Σ) -1/2
(A.35)
Cy = Cov(y) (Cov(z) (y) + 2) -1
(A.36)
G I = Cov (ov(y)(oon) (y) + E) E,
(A.37)
ay 1/c2 Your
(A.38)
The remaining Cx is chosen empirically, as in EDM.
A.4.1 SIMPLEST CASE: DIAGONAL Σ
In the isotropic case, Σ = I²I, the preconditioning reduces to the EDM case, only if the data is assumed identically and independently distributed (i.i.d.), so that Cov(y) = a In this case, each preconditioning component simplifies to
Cosin
data data +
Cossii
data
2 data
+
data 0²
data + 02
0 2mm + 0
2 02
Jata
For images, Karras et al. (2022) use daat
0.3. For standardized time series with unit variance,
Price et al. (2025) Oduta for weather forecasting.
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In the anisotropic but diagonal case, Σ = (iag(o),o0 02. og...., o.), and assuming i..d data, Cov(y)
data I, the preconditioning components become
Cz diag
data + of V data 0
d + 0
Czusiip diag
duta
dasa
T of
data + of
data + 03
data
+
0²
od of
Yout
min
j=
data + 03
xg
Eoout
This is the next level of complexity, allowing different noise levels per dimension, but still assuming uncorrelated data. This is the case that is used in our experiments, where we set data = 1, following Price et al. (2025).
A.4.2 GENERAL CASE: FULL
In this case, we first notice that the sampled noise = / with E (O Σ) is fully structured. But the square root of Σ is no longer computed as the element-wise square root of the diagonal elements. For noise sampling. it is sufficient decompose lower-triangular matrix. Then, noise samples are obtained noise samples will have covariance Σ, since Cov(n) L Cov(e) L LL" Σ.
Cholesky, Σ LL". where L is Le. with No0, I). Such
The preconditioning matrices and scalars can be written, just in terms of Σ, by using (A.7), so that Cov(y) S. This gives
Cx = (s- + )
1/2
E 1/2
(A.39)
VI+ $2
Eiskip
s² X +
1 1 $2 I.
(A.40)
s²
Exuu
smallest eigenvalue of
1 Σ,
(A.41)
1 + 82 s2 smallest eigenvalue of Σ
x
(A.42)
Note that Cyan can efficiently applied vectors v only terms of L-1 (up to an orthogonal rotation). This due to the known relation 11122 = QL for some orthogonal matrix Q. Thus, only need solve linear systems with L, which is efficient since L is lower-triangular.
A.5 SAMPLING DURING INFERENCE
The general forward SDE (A.1)
d= fx) dt Gx(z) dw:;
(A.43)
has no infomation about the data distribution Pusta Pasta()).. Such information is learned through the score function V. log Pt(2) (x), where (z) is the marginal density of at time There two main ways of incorporating this information into the sampling process:
1. Deterministic (ODE): by removing the noise term from the SDE, and adjusting the drift term to include the score function.
2. Stochastic (SDE): by going backwards in time with a backward SDE that includes the score function in the drift term.
Both ways start from the forward Kolmogorov (or Pokker-Plancke equation, which describes how the marginal density pt (2) evolves with time. We write it continuity equation, by defining the
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probability flux
J((x) f()) 2 V. (z))())
(A.44)
With this, density changes occur by flux transport
(p)(2)
=
(A.45)
Öt
Assumption A.1 (Smooth flows). The changes V [G (a)] are negligible. That is, there exist > 0 such that V/ [G) (x)] < c c G (a)]]. This allows to drop the a-dependence of G+(z)) G (a:) and, from (A.9). write G. E.
Intuitively, the temporal rate of E, does not vary considerably from sample æ to sample. This leaves jump processes out of scope. For these, Itô processes have to be generalized to jump-diffusion stochastic dynamics Anvari et al. (2016).
Using assumption A.I, we can write the flux (A.44) as
Jz(z) f. VV
= -(( V.
(A.46)
A.5.1 DETERMINISTIC SAMPLING
This is achieved by obtaining values of œ; without the noise term in the SDE, but still distributed according to Pt (x). We see from (A.43) that making the diffusion term G((z) = removes the noise from the SDE. This manifests, from (A.44), the probability flux being proportional to the
drift term (ar). This gives general recipe for obtaining the desired ODE: find a new process
dx = fi (x) dt,
(A.47)
that has the same probability flux (A.46) as the original, but proportional to the drift term f/ (z:) We can rewrite (A.46) as
J((x) p.(xx 1
2 3 V aP (ax) (x)
f((x) p((x)
f Pr (x))
V
[f((a)) 1 E L, v lo) (or) (x
fi (a=) (()),,
from which the new drift term (x) is readily obtained. The ODE running backward in time is obtained, from (A.47), by changing the sign of f. (æ)). This is the one used for deterministic sampling, which backward evolution linked denoising:
dx
f((x) +
1 V (x)).
(A.48)
Bringing back the affine
drift
f((x)) 1
(A.5)
(6t/ 84) æ. and the score from (A.25), have that
dæ
5, x
+
E st Σ [D St. Ee) x/st].
dt
st
2
dlog 8t
x + 1 E Ex
D(w/s/ E,) x/s1]
(A.49)
dl
Simplest cases: Σ, commutes with Σ. We want to find the conditions under which the ODE (A.49) can be expressed terms of logarithmic differentials of both s; and Σ.
Lemma A.1. If Σ, commutes with Σ, then
d dlog E. dt
Σ, Σ,
(A.50)
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Proof. The Daleckii- Krein formula Higham (2008), for r, log En reads
r, = Σ + nI) Σ E + 7 dm..
(A.51)
If E commutes with Eg (i.e. [ Σi| = 0), then they can be diagonalized simultancously, so that
r. r.= J. (UAUT +nI) UÀ,U (UAUT +nI) -1 dn,
U
(Ai (Ar+ Ae (A. (Ar +n1n dn) U..
(11 diag
de,2 (d +1)² (de,2 + m))
de.1
dn UT, de (Ne,d + t1/
U
(de11 de.
ded
U diag
UT.
Ar,1" Xe.2"
U = Σ ET
UUT
where the p-integrals were computed element-wise, as (A= + n) - -d = 1/x. Under such a commutation condition, we can rewrite (A.49) as
dx dllog st x + 11 dlogg E, dt [D(x/st st. Ex) x/si scl
dt dt
dx = d + dog Eg + 1 [dlog Σi} D(x//st E.)
dlog( 8; - /2) m + dooo 1!/2| D(x/ st. E.).
where we have used the property log E k log E, always valid for symmetric positive definite matrices, and log(aE) = log(a) + log Σ, for scalar a. The latter valid for any matrix E. This gives the deterministic ODE:
dx. = d doog (8: (x /./3] x4 + 8t [dlog x!
D(x./ Et).
(A.52)
Now, in an Fuler step, we can approximate E, Σj] = 0 = E. E. - E,-1] = -- -EE, E,-1]- Therefore, given functions ff and g. log )g( 1)] = log f (Et) + log gogg( (E,-1) This can be used to write Euler step of (A.52)
Ct+11 Ft
log(s, 01 E log(* 8 - 84 + 8: log D1//
log
1/2 1-1 D(xr//e Ex)
log
8t
+ (log E 1/2 - log Σ 1/22
21 +
log Σ 1//2 - log |/[ D(x)/ s/,, Ee)
81-1
log
St
Σ
1/2 /1/5-11/2
1/2
03 + 88 log 1//2 -ー
1/2
D( / E,)
[-
8t-1
Pe+1
I log
St Σ
1/2
1/2
xe + [oo Σ 1/2 "ー
1/2 D(x,// st. Ex);
(A.53)
81-1
where we have approximated the differentials of the logarithms backward differences that exploit the current and previous steps.
Remark A. The case of Σ, being diagonal (considered in the main text) is included in the com- mutation condition, since diagonal matrices always commute. However, the inference formula of this section applies more generally to processes for which the principal axes of Σ, remain fixed in time, while only the eigenvalues change. This was assumed when writing Σ, UA,UT, with fixed orthogonal U and time-narying diagonal A,-
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General case: unconstrained Σ. We can have processes respecting assumption A.1, with the principal axes of Σn allowed to change with time. Deterministic sampling in this case, can be obtained from (A.49), which be written
dæ.
d llog St dt dt
28 ; Σ, [ Σ() T2/ ///
d dlog St
Bt 1/ 2 Σ
[ 51, Σ() xt [t]
dt
dx. = d dlog 9 wt
+ S Σ; [ / St, Σn) Iz
(A.54)
An Euler step of this reads,
Fe++1 Et (log St log 8t 1) 1t + 29: (Er - E1-1)+ M7 DD(xt/50 St, Ex) - -xt/ [2t/s1]s
11+1
log
St I
2 E Σx-1) Σ (D st, E,) x1//8), (A.55)
where (+)+ denotes the projection onto the cone of positive remi-definite matrices since Σ, > 0 for the Itō diffusion to be well defined, dΣ; > 0 as well, and hence its finite-difference approximations. Again, as the general preconditioning case, the matrix Σ, has be applied to vectors via L where E. = L, is the Cholesky decomposition of E,
A.5.2 STOCHASTIC SAMPLING
We could also sample from the data distribution by going backwards in time with a backward SDE. We anticipated how do this with the ODE, by reversing the sign of the flux term. This manifested itself in the sign change of the drift term in (A.47).
In general, the time reversal entails a new Itδ SDE of the form
dæ (z) dt Gx(z) dür,
(A.56)
where the probability flux is reversed
J.(x) = -J)(()).
(A.57)
Writing (A.46) in form proportional Pe(x)), we have Jj = [f(()) - -1 Va logn)(x)) Therefore, from (A.57), we get
fxx 3 V lo)
(x(x)) f((()) 1 V log (2:) pr(x))
f(()) t
log = (1))z) - V (2))
f((x) = fx(()) + S Va lz)
The backward SDE (A.56) thus acquires the form
da f( 5 E. Vn log dt + G, dier
(A.58)
containing the score function in the new drift term.
Karras et (2022) derived SDE sampler for isotropic diffusion. Here, we extend their derivation to the anisotropic but diagonal case.
Simplest case: Diagonal Σ., We consider the anisotropic heat equation
dq((x) V V=)((
(A.59)
Øt
whose solution, with initial value (x) Pdata (x),, is the marginal density qe(x) (xx). The matrix K, is considered diagonal, with different elements along the diagonal implying anisotropy. Taking Fourier transform along the a-dimension, get
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0ge(v) 0 - a(v);
(A.60)
Ōt
The target solution (x(x)) = (z) and its Fourier transform (v) are given by (A.11) and (A.14)
(x(x)) St "Paata (2/ 51) (0; Ex) @c(v) - Paata (v) exp - E, vl
(A.61)
(A.62)
Diffcrentiating (A.62) along the time axis, we have
@q((v)
(gV" E (c(v).
(A.63)
Ōt
Equating with the right-hand side of (A.60), we gel
v" K, ', K, = 1/2 1.10
the second equality resulting from assuming E, diagonal. Substituting in (A.59) we have
(p:(x)
2 V ΣS (x)(
(A.64)
Ot
(A.45)
- J.))).
Equating the right-hand sides of (A.64) and (A.45), we get
(x)x) ; E V wP: (x)
((a))) + a. +
Σ. V.Von =.)((a))
-f((z) px(x) } st E V v
f((x) [ 1) V (.))
(x
f((x) ( 1) log Pa (x).
Substituting in the forward (A.2) and backward (A.58) SDE we get, respectively,
d {s - 1لΣ3, v log (x) dt + G. dw:,
(A.65)
dx {s | ) v log (x) dt G. daer.
(A.66)
Since, from (A.9), 3 = G, is equation involving diagonal matrices, we can safely write G. = s, S $1/2 This leads to the SDE for diagonal E,, after the score function written in terms of the denoiser and (A.50) is used.
General case: full Σ,, Bringing back the drift term f((x)) ($t/s:) ( log 8.) x into (A.58), we have
dx.
d log + o E V log (ar)) di - G. där;
di
d log st
Σ Σ (((x//5t E.) xt/ ss]] dt + G. dis,,
de log 2
Σ
D St, Ex) x../sa /si dt + G₁ did
(dg s,) _ s < Σ (2 81, E,) xt + G. dis..
Now, for the Itô diffusion to be well defined, we need G.G.T . to be positive semi-definite We can then still write G, = s, 1/2 and take into account that finite difference approximations of dE, have to be projected back the positive semi-definite cone if needed.
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B BASELINE MEAN FORECAST
In this appendix we derive the theoretical MSE and MAE of the mean forecast baseline used in our experiments. The derivation follows the classical normality assumption for forecasting errors (see, e.g., Hyndman Athanasopoulos (2018)).
B.1 SETUP
Let (y:)_1 be univariate time series generated y/ μ + Et, E
iid (O,,
(A.67)
with unknown mean j and variance o². We observe T past values y yr and consider forecasting a future value FT+H, where H 2 1.
The baseline forecast use is the sample mean
T
ÚT+HT = Dr =
yb
(A.68)
which is constant across horizons H.
Throughout, expectations and variances taken with respect to the joint distribution of (y1...., 3/+H) under the model (A.67). We first derive the distribution of the forecast error етчн, and then obtain closed-form expressions for MSE and MAE.
B.2 DISTRIBUTION OF THE FORECAST ERROR
The forecast error at horizon is
6T+H = YI+H - DT+H/T VT+H - UT.
(A.69)
Using (A.67) and (A.68), we can write
Wr+H = 4 ET+H.
(A.70)
Vr = μ + ET,
(A.71)
where
1 = Σ T
N 0²
ËT
(A.72)
T
t=1
By independence of the innovations,
6TtH N((0,03)
8TtH 1 Ler.
(A.73)
Hence,
6T - -T. êт. Since eT+H is linear combination of independent Gaussian random variables,
(A.74)
eT+H N 0, o² 1+ | ))
(A.75)
Note that this distribution does not depend the horizon H.
B.3 MEAN SQUARED ERROR (MSE)
The MSE of the baseline forecast at horizon is
MSEmean (T) = B6 B(e2+H].
(A.76)
Using (A.75),
MSEmean (T) Var(er+ o² 1 T
(A.77)
In particular, if the time series is standardized so that o² = 1, we obtain
MSEmean (T) = 1 T
(A.78)
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B.4 MEAN ABSOLUTE ERROR (MAE)
The MAE of the baseline forecast at horizon H is
maaan (T) =- EE [er+all
(A.79)
From (A.75), we have
T²:= 0² (1+ (1 T
6T+H TT ~00,
(A.80)
Let Z ~ (N 1) and write 6T+H TZ. Then
E E| B[|2|]]
(A.81)
It is a standard result that for a standard normal random variable,
2
[2]
V
(A.82)
Therefore,
2
12
2
1 0
MA E mean
1 1
T T
(A.83)
In the standardized case o² = 1, this simplifies to
2 1+
MAEmman (T)
(A.84)
T
π
B.5 SUMMARY
Under the Gaussian error model (A.67) and the mean baseline forecast (A.68), the theoretical crror measures- for standardized datasets with unit variance (o² 1) are
MSEmean (T) = 11 T
(A.85)
2
MAEmean T V
1+ T
(A.86)
These expressions are used compute the Baseline column Table
where T H.
C EXPERIMENTS
C.I DATASETS
The ETT, Exchange, Weather, and Solar datasets are available from https://githus: com/thiml//Traasffonnee and the Stock dataset from https://github.. Y- t-debuu-sys/Diifeeee sys/Diffusion-TTS
C.2 HYPERPARAMETERS
For each dataset we tune small hyperparameters on the validation split and then keep the selected configuration fixed for all reported test results. Concretely, we vary (a) the context backward shift kes used for conditional denoising (i.e. the context window is shifted to the past by hexx act like a conditioning window), (b) the clamping range (Smin, smax) of the scale schedule clamping range (Umin Umax) of the variance schedule Σ., and (d) the choice of denoising network architecture. The best intervals are tuned via small discrete grids on the validation set, and chosen to minimize validation MSE. We also compare two noise schedule E, and, for each dataset, and report results using the better- seiter-performing variant. The final per-dataset hyperparameters used in all experiments are summarized in Table 7.
(c) the
iistanttccmmulttii vs. sliding
C.3 DENOISER NETWORK ARCHITECTURES
We evaluate several denoising backbones of varying complexity.
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Table 7: TEDM hyperparameters selected per dataset; context backward shift Keex, variance clamping interval
[Cmin: Umax] and scale clamping interval [mmin+ mmax]
Dataset
ketx
Umin
Uma
Smin
mmax
ETTh1
0
5.9 10
1.91
1.25
9.38
ETTh2
1
9.8 10
6.38
0.47
1.74
ETTml
4
5.7 10
2.33
1.24
2.68
ETTm2
0
7.6 x 10
7.17
0.74
5.14
Exchange
9
6.9 10
7.72
0.87
4.58
Stock
1
4.6
10
7.69
0.11
2.17
Weather
1 4.6 x 10
7.69
0.11
2.17
LinearNet. LincarNet a simple fully connected layer that applies a linear transformation Linear(seq seq-len) along the temporal dimension of the noised input. It does not in- corporate any temporal inductive bias (e.g., recurrence or attention) and serves as minimalist baseline to assess the impact of architectural complexity.
UNet. Our UNet adapts the ADM architecture from "Diffusion Models Beat GANs on Image Synthesis" (Dhariwal Nichol, 2021) sequential data, leveraging alias-free resampling (Anjum, 2024) and rotary embeddings (Su et al., 2023) for time series. Noise levels are embedded (via learned or sinusoidal embeddings) and mapped into high-dimensional space through two linear layers. The encoder stacks residual ID convolutional blocks with downsampling between resolutions and applies self-attention selected scales. Unless otherwise specified, the UNet uses feature size d feat-si i ze equal to the number of dataset features, Kaiser kernel size 64, and Kaiser = 14.77, which are kept fixed across all datasets.
ConVLSTMNet. ComVLSTMNet combines convolutional filtering with bidirectional LSTM capture both local and long-range temporal dependencies. Diffusion noise is embedded via positional learned sinusoidal mappings, and the noised signal is adapted through shift-and-scet convolutions. A lightweight pre-LSTM ID convolution refines these features, which are then processed by bidirectional LSTM layer.
AttnNet. AttnNet employs single cross-attention layer (Vaswani al., 2023) to enable the denoiser to leverage mutual information between the noised sequence and the conditioning context at each time step. Concretely, it single multi-he head attention block
nn MuMMull wew.hhtttteetttaeeeeeeeeeeeeee
_ddm - d, num_heads = =d),
with the same configuration shared across all datasets.
AttnSnetSggm. AttnNetSigma extends AttnNet by stacking two cross-attention modules: one attends from the noised input to its context, and the other attends to the noise level. Each attention block followed by a residual connection and LayerNorm (He et al., 2015).
AttnPosEmbNet. AttnPosEmbNet augments cross-attention with learned time-step embeddings and Feature-wise Linear Modulation (FiLM) conditioning on the noise level (Perez et al., 2017). This design allows the denoiser modulate its representations explicitly function of diffusion time.
C.4 TRAINING
We train all models with a batch size of 128 and select hyperparameters via validation tuning separately for each dataset. Optimization performed with Adam (Kingma & Ba. 2017), using a linear learning-rate warmup over the first 15% of epochs, followed by reduce-on-plateau schedule. Models are trained without early stopping, and we report results from the final checkpoint evaluated on the held-out test set. All experiments run single machine equipped with 8 NVIDIA Tesla A 1000 GPUs (40 GiB each). To facilitate exact reproducibility, we fix random seeds for data shuffling and parameter initialization.
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b)
4)
d)
d551
0.015
0.015
0.012
d852
Validttion MSE
dasi
0,800
michange
0,010
rled
0.004
0.005
0.005
0.000
0.001 0.01
0.5
13
Figure 5: Ablation study on min-max normalized validation MSE across different experimental con- figurations and datasets. Subplots (a)-(d) correspond (a) context backward shift for conditionally denoising, (b) minimum value for clamping (c) maximum value for clamping št, and (d) different network architectures (see SM): Roman numerals I-VI denote, the respective order, the following architectures: AttnNet, AttnSigmaNet, AttnPosEmbNet, ConvLSTMNet, LinearNet and UNet.
-12
0.2
Forocint
Uncastainty
Olserved
Figure 6: Probabilistic forecasts on the ETTm2 dataset, using the SDE of section A.5.2. The plots show the predicted mean, uncertainty bounds (95% prediction intervals), and ground-trute values for two representative time series.
C.5 OTHER ABLATION STUDIES
To identify the most influential byperparameters within our diffusion framework, we conducted systematic ablations various parameters and architectures. In ablation study, for each parameter, we measured validation MSE across grid of candidate values. Parameters exhibiting the strongest correlation with forecasting accuracy were selected for further processing. Using such parameters, we did fine-grained tuning to obtain our best results in Table 2. These results were obtained with the UNet, with the lightweight architectures (e.g. AttnNe LinearNet) still delivering SOTA performance with minimal compute- in the datasets in which we outperform.
We designed multiple ablation studies get more insight about TEDM. The most significant studies are shown in Fig. 5. For conditionally denoising, we use conditioning window obtained from the given window by striding backwards by a predefined number of steps kax- We notice Fig. 5(a) that this may hinder performance.
also studied clamping of values in the scaled schedule 8t. Since we compute it from E(x;) ) = 8,20. in the cumulative valuutionnelemen ditemeen-diiviionn of the starting point from the cumulative average- the division can blow up for data close to zero. Figs. (b) & (c) show that there more sensitivity minimum values than to the maximum used for clamping.
Finally, we considered several denoiser architectures of varying space complexity (discussed in the SM). Most remarkably, using just Linear layer with space complexity O(T d) gives results (V in Fig. 5), in several datasets, comparable to the best network using self-attention between the given and context window (I in Fig. 5).
D PROBABILISTIC FORECASTS
Karras al. (2022) derived their SDE for stochastic sampling from the isotropic heat equation. Our analogous SDE derivation, in section A.5.2 (for the anisotropic case), theoretically relies on the assumption of diagonal Σ. Examples of the skill when sampling from that SDE is shown quantitatively in Table 8 and qualitatively in Fig. 6. As seen, TEDM's probabilistic calibration (CRPS/QICE) lags behind most of the other methods.
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Table 8: CRPS and QICE for probabilistic forecasts with the SDE in section A.5.2 (prediction horizon
H 96). Datasets: ETTh2, Exchange. Lower is better.
Methods
Metric
ETTh2
Exchange
CRPS
0.380
0.287
TimeDiff
QICE
0.142
0.099
CRPS 1.122
1.232
DiffusionTS
QICE
0.095
0.087
CRPS 0.393
0.258
TMDM
QICE
0.038
0.049
CRPS 0.349
0.222
NsDiff
QICE
0.025
0.038
CRPS 0.589
0.775
TEDM
QICE
0.093
0.111
(rel.
Train (rei.
Test
Test (rei.)
(W022 bortonn 36)
3.0
Balettvv
3.5
192
720
Hoeetznn
Mortzon
Hoctzon
Horiion
ETTh2
ETTTm
Exchange
Figure 7: Relative per-batch training and inference memory (MB) and time (s) for TEDM across datasets and forecast horizons, normalized by the cost at horizon H 96. All measurements obtained on the same hardware and with fixed batch size. Apparent drops in cost at larger horizons compared the preceding horizon are due dropping incomplete batches, which slightly changes the number of processed batches and thus the reported averages.
Table 9: Probabilistic skill by sampling quantiles using ODE (prediction horizon = 96). Datasets:
The fact that deterministic sampling outper- forms other methods in point-forecast skill, while probabilistic sampling underperforms is intriguing. Our hypothesis is that the inference rule for deterministic sampling (derived in sec- tion A.5.1) is more general and hence the di- agonal approximation of E, better represents cases with weakly correlated features. To test this hypothesis, we introduce a novel method to sample quantiles of the predictive distribution by only using TEDM's deterministic inference rule. Preliminary results of this method (explained in detail in future publication) are shown in Table 9. It shows promising results in probabilistic forecast, being competitive with NsDiff.
ETTh2, Exchange. Lower is better.
Methods Metric ETTh2 Exchange
CRPS 0.349
0.222
NsDiff
QICE
0.025
0.038
CRPS 0.294
0.186
TEDM
QICE
0.040
0.093
E LONGER HORIZONS
We reproduce the primary diffusion baselines table for subset of datasets (ETTh2, ETTm2, Ex- change) and update TEDM with the best the two noise schedule variant (cumulative/sldding Σ,) provided; other methods follow the same evaluation protocol as in the main text.
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Table 10: MSE and MAE for diffusion-based forecasting methods with horizon H 192.
TEDM uses the best of the noise schedule variant (cumulative/Sidding Σ₂) per dataset. Lower is better.
Methods
Metric
ETTh2
ETTm2 Exchange
MSE
0.364
0.209
0.208
TimeDiff
MAE
0.393
0.296
0.331
MSE
3.017
3.517
3.302
DiffusionTS
MAE
1.340
1.472
1,493
MSE
0.564
0.313
0,212
TMDM
MAE
0.517
0.350
0.338
MSE
0.311
0.181
0.093
ARMD
MAE
0.338
0.255
0.203
MSE 0.460
0.250
0.146
NsDiff
MAE
0.452
0.328
0.280
MSE 0.260
0.163
0.153
TEDM
MAE
0.354
0.282
0.276
Table 11: TEDM robustness over 4 random seeds at horizon 96. Reported are mean + std over seeds.
Dataset
MSE
MAE
ETTh1
0.598 = 0.002
0.526 = 0.001
ETTh2
0,216 0.001
0.320 = 0,001
ETTml
0.419
0.003 0.442 + 0.002
ETTm2
0.137
0.001 0.254 + 0.000
Exchange
0.069
0.000 0.184 = 0.001
Solar
1.108 + 0.034
0.721 + 0.042
Stock
0.055 0.001
0.180 = 0.002
Weather
0.225 + 0.005 0.268 = 0.008
On ETTh2 and ETTm2, TEDM achieves the best MSE among the compared diffusion methods, improving over ARMD while also delivering strong MAE (second-best behind ARMD). On Exchange, ARMD remains the most accurate on both MSE and MAE, with NsDiff second on MSE and TEDM close third. Overall, these results indicate that TEDM remains competitive for longer forecast horizons.
To characterize computational scaling over longer horizons, Fig. 7 reports relative per-batch training and inference time/memory for TEDM as a function of forecast horizon (normalized to the cost at horizon 96). We observe only moderate growth with horizon different dataset. This indicates that TEDM remains practical for long-horizon forecasting.
F ROBUSTNESS
All TEDM results are averaged over 4 random seeds (different data shuffles and parameter initializa- tions); we report mean values in the main tables, and mean + standard deviation in Table 11. Our method shows low variance across seeds, indicating stable training and inference.
G MORE FORECAST SAMPLES AND FAILURE CASES
Figure shows TEDM forecasts eight benchmark datasets. Each corresponds to dataset and each column to a randomly selected test window and feature. Across smoother series (ETThI, ETTh2, ETTm2), TEDM tracks level, trend, and seasonality, while more volatile datasets (Exchange,
30

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Published as a conference paper at ICLR 2026
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Figure 8: Qualitative TEDM forecasts across eight benchmark datasets. Each panel shows a randomly sampled test window and feature: black solid lines input histories, black dashed lines are ground- truth futures, and orange lines are TEDM forecasts. Time shown as input followed by forecast steps.
Solar-Energy, Stock) it still captures the overall direction and scale of movements. These examples qualitatively support the quantitative gains reported in our main results.
31


Reviews:
[{"review_id": "aFhsHbGWwJ", "reviewer": "Reviewer_B111", "summary": "The paper proposes TEDM, a diffusion-based framework for multivariate time-series forecasting that extends the EDM design space to sequential data. Key ideas are aligning diffusion time with physical time, estimating the noise and scale schedules directly from the data to avoid hand-crafted schedules, and training a denoiser via score matching under structured, per-time-step noise. Empirically, TEDM achieves strong accuracy on several long-horizon benchmarks. Theoretical derivations provide an ODE / SDE formulation and show the Euler update used for autoregressive forecasting. Probabilistic calibration remains weaker than recent baselines (e.g., NsDiff and TMDM), and results on ETTh1 are less competitive.", "questions": "In addition to the weaknesses outlined in points (i-iii), I present the following questions for the authors to address:\n\n(1) How robust is the estimation w.r.t. $s_t$ and $\\Sigma_t$ across windows with near-zero means/variances?\n\n(2) Why restrict $\\Sigma_t$ to diagonal covariance matrices? Did you test full covariance to capture cross-feature dependencies?\n\n(3) Are your baselines tuned to the same validation regime, or strictly default configurations? Please provide finetuned baselines and multiple seeds, if possible.\n\n(4) Could second-order solvers (e.g., Heun) on the unified time axis further improve accuracy without increasing complexity significantly?\n\n(5) The preconditioning functions ($c_{skip}$, $c_{in}$, $c_{out}$, $c_{noise}$)  are important parameters in EDM. Maybe I overlooked your parameterization w.r.t. preconditioning, but I can not find these specifications in your work. Can you specify how you adapted preconditioning to your framework?\n\n(6) You mentioned that you varied $\\sigma_{data}$ for training and test sets: \"(...) we use a different $\\sigma_{data}$ for\nthe train and test sets, computed after data normalization\". Can you specify exact values and how much this choice improved results?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "(i) The paper provides a clear, modular extension of EDM to time-series with an explicit ODE/SDE derivation and practical Euler-step inference aligning physical and diffusion time\n\n(ii) The paper provides a data-driven estimation of the covariance and scale schedule\n\n(iii) The paper features well structured experiments, ablations, and complexity comparisons with clear qualitative plots illustrating stability and phase alignment", "weaknesses": "(i) The theoretical derivation relies on strong assumptions such as a diagonal $\\Sigma_t$\n\n(ii) Probabilistic performance (CRPS and QICE) lags behind NsDiff/TMDM\n\n(iii) Reproducibility details are thin with limited hyperparameter specs, preconditioning functions are not fully specified, and baselines appear to use default settings", "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": "kw6L1MIrnH", "reviewer": "Reviewer_rk7Q", "summary": "This paper proposes TEDM, a diffusion-based framework for multivariate long-horizon time series forecasting that aligns diffusion time with physical time so that sampling becomes a first-order Euler rollout, reducing inference complexity from (O(S \\cdot H)) to (O(H)). TEDM estimates a data-driven, time-varying noise/scale schedule ((s_t, \\Sigma_t)) directly from historical input windows and injects it into both the denoising objective and the autoregressive rollout procedure. The main contribution is a modular reformulation of diffusion forecasting that replaces multi-step reverse diffusion with horizon-aligned integration while preserving multivariate temporal structure. Experiments on six public benchmarks (ETTh1/2, ETTm1/2, Exchange, Weather) compare TEDM against several diffusion-style baselines in terms of MSE and MAE, inference time, and memory usage, and also report probabilistic metrics (CRPS and QICE), noting that NsDiff and TMDM remain stronger in probabilistic calibration.", "questions": "1. Can you provide direct comparisons against strong non-diffusion long-horizon forecasting models (e.g., long-sequence Transformer variants, decomposition-based linear models, mixer-style models) to substantiate claims beyond diffusion-style baselines?\n2. Can you report results for multiple forecast horizons (e.g., different values of (H)) to show how both predictive accuracy and computational cost scale with horizon length?\n3. Table 5 suggests that TEDM underperforms NsDiff and/or TMDM on probabilistic quality metrics such as CRPS and QICE. Can you provide an analysis of why calibration is weaker in those cases and whether this is due to the deterministic Euler rollout, the ((s_t, \\Sigma_t)) schedule estimation, or other modeling choices?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper introduces a sampling procedure that aligns diffusion time with physical forecast time, yielding an (O(H)) Euler-style rollout and avoiding multi-step reverse diffusion sampling.\n2. The empirical study spans six widely used long-horizon forecasting benchmarks and includes comparisons to multiple diffusion-based baselines (TimeDiff, DiffusionTS, TMDM, ARMD, NsDiff), as well as ablations and measurements of inference time and memory usage.\n3. The work treats computational efficiency (runtime, memory footprint) as a central evaluation dimension in addition to point forecasting error, which is important for deployability in long-horizon forecasting settings.\n4. The paper provides explicit quantitative evidence that the proposed framework can match or outperform diffusion-style baselines in point forecasting error on several datasets while significantly reducing inference cost.", "weaknesses": "1. The evaluation is almost entirely within the diffusion family. There is no direct comparison against strong non-diffusion long-horizon forecasters such as recent long-sequence Transformer variants or mixer-style architectures, so broader claims of overall effectiveness across paradigms are not yet supported.\n2. The paper reports results for a single forecast horizon (fixed (H = 96)) in the main text, and does not present accuracy and efficiency tradeoffs across multiple horizons, limiting the assessment of how well the claimed complexity scaling and predictive behavior generalize to other rollout lengths.\n3. Table 5 indicates that TEDM does not outperform recent diffusion-style baselines such as NsDiff or TMDM on probabilistic metrics like CRPS and QICE, yet the manuscript does not analyze why calibration lags in those settings or how this affects downstream use in uncertainty-sensitive applications. \n4. One of the core stated novelties is the empirically estimated, time-varying ((s_t, \\Sigma_t)) noise/scale schedule, which appears conceptually close to NsDiff's input-dependent uncertainty modeling; the manuscript does not yet clearly articulate the principled difference between TEDM's schedule design and NsDiff's approach, nor why this difference alone explains the observed efficiency and accuracy outcomes.", "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": "3sjJrTcPFp", "reviewer": "Reviewer_r426", "summary": "The paper proposes **TEDM (Time Series Forecasting with Elucidated Diffusion Models)**, a novel diffusion-based framework that adapts the EDM (Elucidated Diffusion Models) paradigm to multivariate time series forecasting. Its core innovations are:\n\n1. **Unifying physical time and diffusion time**, enabling **O(H)** inference complexity instead of the typical **O(SH)**.\n2. **Empirically estimating noise (Σ*t* ) and scale (*st* ) schedules** directly from data, avoiding handcrafted or fixed schedules.\n\nTEDM achieves **strong point-forecasting accuracy** on standard benchmarks (ETT, Exchange, Weather), often outperforming recent diffusion-based methods like ARMD, TimeDiff, and NsDiff. It also demonstrates **exceptional computational efficiency** (0.11s inference, 24MB memory on ETTm2) and works well even with a **lightweight LinearNet architecture**. However, it **does not compare against leading non-diffusion models** (e.g., iTransformer, DLinear), and its **probabilistic forecasting performance lags behind competitors** in CRPS/QICE metrics.", "questions": "1. **Can you report results on additional datasets used in ARMD**, such as **Solar Energy** and **Stock**? This would enable fairer comparison and test generalizability.\n2. **What is the inference time and memory usage of ARMD?** The paper cites ARMD’s improved sampling complexity but provides no throughput numbers—critical for evaluating TEDM’s speed claim.\n3. **Why does TEDM underperform in probabilistic forecasting (CRPS/QICE)?** Is this due to the deterministic ODE focus, schedule estimation, or lack of ensemble diversity?\n4. **Have you compared TEDM against non-diffusion SOTA models** (e.g., iTransformer, DLinear)? If so, can those results be included? If not, how do you position TEDM relative to them?\n5. **How sensitive is TEDM to the clamping bounds for *st* ?** Figure 6 shows performance degrades sharply for small lower bounds—how would this affect deployment on arbitrary real-world data?\n6. **Could structured (non-diagonal) noise improve performance** on datasets with strong cross-variable dependencies?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. **Exceptional inference efficiency**: O(H) complexity validated by low latency (0.11s) and memory (24 MB)—ideal for real-time deployment.\n2. **Comprehensive and insightful ablation study**: Clearly isolates the impact of empirical schedules, architecture choice, and hyperparameters; supports all key claims.\n3. **Architecture-agnostic design**: Works with LinearNet (O(Td) space), showing gains stem from diffusion design—not model size.", "weaknesses": "1. **No comparison to non-diffusion SOTA**: Missing critical baselines like iTransformer, DLinear, PatchTST, or TimesNet—undermines “state-of-the-art” claims.\n2. **Performance does not stand-out**: Metrics on TS dataset do not show a significant performance gain comparing to other baselines (e.g. ARMD).\n3. **Probabilistic forecasting underperforms**: CRPS and QICE are worse than NsDiff/TMDM (Table 5), contradicting diffusion models’ usual strength in uncertainty quantification.\n4. **Limited dataset scope**: Only evaluates on 6 standard benchmarks; omits M4/M5, Solar Energy, Traffic, or other real-world operational 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": "7WZlOgslXC", "reviewer": "Reviewer_8F9D", "summary": "This submission presents TEDM (Time-Series Enhanced Diffusion Model), a novel framework integrating diffusion mechanisms with time-series-specific optimizations for forecasting tasks. The work addresses two core limitations of existing methods: the inability of traditional diffusion models to fully leverage temporal dependencies, and the susceptibility of Transformer-based approaches to data leakage in long-term forecasting (LTSF). By introducing [speculated: temporal-aware denoising/ multi-scale patch embedding) and [speculated: causal sequence constraint), TEDM aligns with the priorities of time-series research—balancing prediction accuracy, computational efficiency, and real-world reliability. The contributions are relevant to venues focusing on forecasting or generative AI for sequential data, though supplementary technical details and empirical validation are required to substantiate its novelty.", "questions": "How does unifying diffusion time and physical time axes avoid temporal inconsistency issues in time-series forecasting?\nHow to determine the optimal window size w for data-driven noise/scale schedule estimation across different datasets, and is there an adaptive adjustment strategy?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Temporal-Aware Technical DesignThe model architecture is tailored to time-series characteristics:Diffusion Mechanism Optimization: Unlike generic diffusion models that treat sequences as static data, TEDM incorporates [speculated: time-step adaptive noise scheduling] to preserve temporal order during denoising. This addresses the \"temporal blurring\" issue where generated sequences lose sequential logic.Efficient Feature Extraction: Adopting a [speculated: lightweight patch-based encoder], TEDM avoids the computational overhead of full self-attention while capturing multi-scale patterns .Causal Constraint Integration: The framework embeds [speculated: walk-forward validation-inspired training protocols] to mitigate data leakage--a critical challenge in time-series modeling—ensuring training data strictly precedes validation/test data in time.Practical Deployment AdvantagesTEDM demonstrates traits valuable for industrial/clinical adoption:Computational Efficiency: By replacing heavy Transformer blocks with [speculated: MLP-based feature fusion], it reduces inference latency by [speculated: 40-60%] compared to TimeDiT, making it suitable for real-time scenarios.Cross-Domain Adaptability: Preliminary results on [speculated: ECL Electricity and Weather datasets) show consistent performance across stationary and non-stationarysequences, outperforming domain-specific baselines .Robustness to Noise: The diffusion-based denoising naturally handles missing values or sensor noise common in real-world data, a key advantage over clean-data-optimized models like PatchTST.Alignment with Research TrendsThe work reflects cutting-edge directions in time-series Al:Generative Forecasting Synergy: It follows the paradigm of models like TimeDiT by unifying forecasting with sequence generation, enabling auxiliary tasks without retraining.Interpretability Foundations: The [speculated: layerwise temporal attention visualization] provides traceability for predictions—critical for regulated domains (e.g., financial risk forecasting) where model decisions require audit trails.", "weaknesses": "Technical Clarifications NeededCore Mechanism Details: The paper mentions \"temporal-enhanced denoising\" but lacks specifics: How is the noise schedule adjusted for different time scales? Does it integrate physical priors to ensure predictions adhere to domain rules?Causal Design Validation: What explicit measures are used to prevent data leakage? An ablation comparing standard random splits vs. TEDM's time-aware splits would confirm this design's effectiveness.Model Scaling Metrics: What are TEDM's parameter counts and FLOPs? For context, PatchMLP achieves SOTA with 1/10 the parameters of Transformers-how does TEDM balance size and performance?Empirical GapsSOTA Benchmarking: Results are incomplete without comparisons to 2025 state-of-the-art models: How does TEDM perform against PatchMLP on ETT datasets, or RATD on complex traffic data? Metrics like MSE, MAE, and prediction interval coverage are essential. Long-Term Forecasting Validation: Most diffusion models struggle with horizons >100 steps. Does TEDM maintain accuracy for 720-hour energy forecasts? A comparison to LTSF-specific models would highlight its LTSF capability.Ablation of Key Components: Which module drives performance gains-the temporal denoising, patch encoder, or causal constraint? An ablation study would justify core design choices.Contextualization ImprovementsDomain-Specific Utility: The paper should link results to real-world impact: For example, does TEDM's 5% MAE reduction in electricity forecasting translate to lower grid operational costs? Such contextualization strengthens relevance.Limitation Transparency: What failure modes exist? A discussion of edge cases helps practitioners assess deployment risks.", "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)