Diffusion Models in Time-Series Forecasting

• Diffusion models in time-series forecasting are generative models that iteratively denoise data to capture complex, multimodal temporal dependencies.
• They employ forward noising processes and reverse Gaussian denoisers parameterized by neural networks to achieve probabilistic and point forecasting.
• Advanced architectures integrate conditional, self-guided, and multimodal techniques to enhance uncertainty quantification and forecast accuracy.

Diffusion models in time-series forecasting comprise a class of generative models that capture complex, multimodal temporal dependencies by learning to simulate the evolution of data through a structured sequence of stochastic perturbations and denoising steps. By inverting tractable "noising" processes (typically Gaussian Markov chains or Itô SDEs) via neural parameterized denoising, these models generate samples from intricate time-series distributions. Their flexibility enables state-of-the-art probabilistic and point forecasting, robust uncertainty quantification, and applicability to multivariate, high-dimensional, and even multimodal settings.

1. Mathematical Foundations and Diffusion Formulation

Diffusion models for time-series forecasting are instantiated by mapping a predictive target (e.g., future values $y_0 \in \mathbb{R}^{H \times D}$) to high-entropy latent representations by a forward noising process composed of $K$ discrete Markovian Gaussian steps:

$$q(y_{1:K}|y_0) = \prod_{k=1}^K \mathcal{N}(y_k; \sqrt{1-\beta_k} y_{k-1}, \beta_k I)$$

with closed-form

$$y_k = \sqrt{\bar{\alpha}_k} y_0 + \sqrt{1-\bar{\alpha}_k} \epsilon, \quad \epsilon \sim \mathcal{N}(0,I), \quad \bar{\alpha}_k = \prod_{s=1}^k (1-\beta_s)$$

The reverse process is parameterized as a sequence of Gaussian denoisers, where at each step a neural network $\epsilon_\theta$ or $x_\theta$ predicts the added noise or clean data, and the conditional mean for each reverse transition is given by

$$p_\theta(y_{k-1}|y_k, x) = \mathcal{N}(y_{k-1}; \mu_\theta(y_k, x, k), \tilde{\beta}_k I)$$

Training employs simplified noise-prediction objectives (Ho et al., 2020) by minimizing the $L_2$ deviation between sampled $\epsilon$ and $\epsilon_\theta$ at random diffusion steps and conditioning contexts, resulting in losses of the generic form:

$$\mathbb{E}_{y_0,x,\epsilon}\Big[\| \epsilon - \epsilon_\theta( \sqrt{ \bar{\alpha}_k } y_0 + \sqrt{ 1-\bar{\alpha}_k } \epsilon, x, k ) \|^2 \Big]$$

Various architectural innovations extend this paradigm to handle channel dependencies, multimodal signals, and non-stationary contextual dynamics (2410.02168, Shen et al., 2023, Ding et al., 24 Nov 2025).

2. Conditional and Self-Guided Diffusion Forecasting

Conditional diffusion models for time series forecast future values given a history window $x \in \mathbb{R}^{L \times D}$. Information integration occurs via embedding the history, covariates, or exogenous variables into the denoising network at every reverse step. Feature-centric conditioning, as in TimeGrad or CSDI, leverages recurrent or transformer state summarization of historical observations; channel-aware architectures (e.g., CCDM’s CiDM + DiT) address intra- and cross-channel coupling at scale.

Self-guidance mechanisms enable unconditional diffusion models to be used for forecasting via test-time conditioning. For example, TSDiff enables observation-based gradients (score functions) to be used for post-hoc guidance, either via mean-square error or quantile-driven loss gradients injected at each reverse step, without altering the original network or requiring re-training (Kollovieh et al., 2023). This approach generalizes to refine or synthesize samples.

Contrastive and reference-based conditions add further regularization: CCDM maximizes mutual information between history and forecast via denoising-based InfoNCE objectives, while retrieval-augmented schemes (RATD) retrieve similar historical trajectories from a database to inject reference targets through Attention during the denoising path (Liu et al., 24 Oct 2024).

3. Advanced Architectures and Extensions

Diffusion backbones have advanced substantially:

• Channel-aware architectures: CCDM integrates parallel channel-wise dense networks (CiDM) and channel-mixing transformers (DiT) for scalable multivariate denoising, supporting hundreds of series with strong cross-dependency learning (2410.02168).
• Multi-granularity conditioning: MG-TSD synchronizes denoising diffusion at multiple time-scale resolutions, aligning each with coarsened temporally-averaged targets, to stabilize learning and preserve both high and low-frequency structure (Fan et al., 9 Mar 2024).
• Continuous-time and non-autoregressive models: SDE-based schemes (ScoreGrad) and deterministic ARMD/Brownian bridge formulations reduce boundary noise and improve stability while covering potentially non-stationary regimes (Gao et al., 12 Dec 2024, Yang et al., 7 Nov 2024).
• Latent and multimodal diffusions: LDM4TS translates time series into visual encodings, denoises in image-latent space, and fuses cross-modal features for vision-enhanced forecasting. MCD-TSF incorporates external text and timestamp modalities via transformer-based cross-attention and classifier-free guidance for multimodal uncertainty-aware predictions (Ruan et al., 16 Feb 2025, Su et al., 28 Apr 2025).

Architecture	Conditioning Modality	Denoising Design
CCDM (2410.02168)	Channel-aware history	CiDM + DiT hybrid
TimeDiff (Shen et al., 2023)	Mixup+AR history	Conv stack + AR initializer
MG-TSD (Fan et al., 9 Mar 2024)	Multi-res history	Simultaneous GRUs/U-Nets
ARMD (Gao et al., 12 Dec 2024)	Sliding-window sequence	Distance-based linear
LDM4TS (Ruan et al., 16 Feb 2025)	Vision & freq/text	Latent U-Net + cross-modal
MCD-TSF (Su et al., 28 Apr 2025)	Text/timestamp/history	Multimodal transformer

4. Theoretical Insights and Empirical Performance

Theoretical developments have demonstrated that standard denoising losses yield tractable lower bounds on predictive mutual information between history and forecast, and contrastive objectives can further regularize denoisers against out-of-distribution error by calibrating forecasts on negative (implausible) paths (2410.02168). For models incorporating expressive priors or interpolation (e.g., ARMD, S²DBM), deterministic or Brownian-bridge chains align diffusion steps with the desired temporal evolution, improving both stability and mean squared error minimization (Yang et al., 7 Nov 2024, Gao et al., 12 Dec 2024).

Empirically, the strongest diffusion forecasters (CCDM, MG-TSD, TimeDiff, S²DBM, SimDiff) achieve best or second-best performance across benchmarks in MSE/CRPS/MAE on canonical datasets (ETTh1/2, Traffic, Exchange, Electricity, Solar, Weather), with relative improvements in error metrics ranging from 9%–47% over prior state-of-the-art baselines in challenging settings:

• CCDM attains best MSE in 66.7% and best CRPS in 83.3% of scenarios; ablations confirm large degradations (+8.3% MSE, +26.1% CRPS) when removing contrastive losses or channel mixing (2410.02168).
• SimDiff, with a unified transformer backbone and median-of-means ensemble, reduces point-wise MSE by 8.3% over previous diffusion models (Ding et al., 24 Nov 2025).
• MG-TSD’s multi-scale guidance secures the lowest probabilistic (CRPS) and deterministic errors on six multivariate datasets (Fan et al., 9 Mar 2024).
• S²DBM's Brownian-bridge yields deterministic forecasts with reduced variance, outperforming both generative and direct regression baselines in point and probabilistic forecasting (Yang et al., 7 Nov 2024).

5. Contemporary Challenges and Developments

Current limitations center on:

• Inference efficiency: Standard iterative reverse diffusion (100–1000 steps) can be slow; accelerated solvers (e.g., DDIM, DPM-Solver), deterministic bridges (S²DBM, ARMD), or tailored ensembling (SimDiff) partially mitigate but do not eliminate this bottleneck (Yang et al., 7 Nov 2024, Gao et al., 12 Dec 2024, Ding et al., 24 Nov 2025).
• Scalability: Some transformer-based architectures have quadratic complexity in the number of channels or horizon, presenting challenges for very high-dimensional multivariate series (2410.02168, Shen et al., 2023).
• Distributional robustness: Adaptation to non-stationary or distributionally-shifted test data requires normalization-independent strategies (SimDiff) or auxiliary calibration (CSDI + conformal inference) (Ding et al., 24 Nov 2025, Pearson et al., 10 Jun 2025).
• Uncertainty calibration: While sample ensembles quantify predictive uncertainty, precise separation of aleatoric and epistemic components remains an open problem.

Recent progress addresses guidance for semantic and covariate fidelity (SemGuide), retrieval augmentation (RATD), and multimodal integration (LDM4TS, MCD-TSF). These trends reflect increasing attention to distributional robustness, interpretability, and the inclusion of diverse auxiliary signals (Ding et al., 3 Aug 2025, Liu et al., 24 Oct 2024, Ruan et al., 16 Feb 2025, Su et al., 28 Apr 2025).

6. Taxonomy and Outlook

Diffusion model frameworks for time-series are now comprehensively categorized as follows (Su et al., 19 Jul 2025, Meijer et al., 5 Jan 2024):

• Feature-centric: Direct conditioning on historical/future windows and covariates (TimeGrad, CSDI, TimeDiff, CCDM).
• Diffusion-centric: Incorporation of time-series priors or prior-informed kernels (TMDM, S²DBM, ARMD, MG-TSD).
• Multimodal and retrieval-based: Vision/text/exogenous fusion (LDM4TS, MCD-TSF, RATD).
• Guidance and regularization: Denoising-based contrastive, classifier-free, or self-guided (CCDM, TSDiff, SemGuide).
• Latent structure and decomposition: Explicit factor/scale (Diffusion-Index models, latent diffusion, multi-granularity, or stochastic prior: StochDiff).

Future research avenues highlight fast (single-step) sampling, theory-grounded calibration, large-scale foundation time-series models, and principled robustness to both covariate and temporal drift (Su et al., 19 Jul 2025, Yang et al., 7 Nov 2024, Ding et al., 24 Nov 2025, 2410.02168).

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References: (2410.02168, Shen et al., 2023, Ding et al., 24 Nov 2025, Fan et al., 9 Mar 2024, Gao et al., 12 Dec 2024, Yang et al., 7 Nov 2024, Su et al., 19 Jul 2025, Meijer et al., 5 Jan 2024, Kollovieh et al., 2023, Su et al., 28 Apr 2025, Ding et al., 3 Aug 2025, Ruan et al., 16 Feb 2025, Liu et al., 5 Jun 2024, Zarifis et al., 19 Mar 2025, Pearson et al., 10 Jun 2025, Liu et al., 24 Oct 2024)