Two Time Axes: Five Years of Diffusion Models and Probabilistic Time Series Forecasting (2021–2026)

1. Introduction: Problem and Criteria¶

The task statement for multivariate probabilistic time series forecasting is short: given history $\mathbf{x}^0_{1:t_0-1}$ of a $D$-dimensional sequence and covariates, output the joint predictive distribution over several future steps—not $D$ marginals, and certainly not $D$ point forecasts. When $D$ reaches hundreds to thousands (power load, traffic cross-sections, portfolio-level risk factors), the entire difficulty concentrates in one place: expressivity of the emission distribution.

Before 2021, three families of approaches each failed in a predictable way. Full-covariance Gaussians require $O(D^2)$ parameters and $O(D^3)$ likelihood computation, and express only second-order dependence; low-rank copulas (the GP-Copula line) trade expressivity for tractability; normalizing flows offer tractable likelihoods but are constrained by the structure of Jacobian determinants—and, as a geometric statement, continuous invertible maps cannot tear apart the support, so they assign spurious density between disconnected modes [1, 13].

TimeGrad [1] answered by adapting the denoising diffusion model (DDPM [2, 3]) that had just matured in images into a conditional emission head for sequence models. Five years later, TEDM at ICLR 2026 [29] has reduced sampling complexity to linear in the forecast horizon, and SimDiff at AAAI 2026 [30] puts diffusion models in direct competition with regression methods on the most conservative criterion—point forecast mean squared error. This note does not aim to exhaust the literature of those five years (see the systematic survey by Su et al. [34]); instead it extracts two threads and strings together papers that "moved positions in the coordinate system":

Thread one (vertical): The literature's evolution replays the logical structure of theory. The mathematics of score-based generative modeling is a one-way street: first score functions bypass the partition function, then Langevin sampling cashes out the score; first the discrete chain is written as discretization of an SDE, then Anderson's reverse-time equation follows; first the probability-flow ODE with matching marginals, then flow matching turns the reference measure into a design variable; finally at the distribution layer of Fokker–Planck, all the above objects become different projections of the same equation. We will see that representative papers from 2021–2026 instantiate this sequence step by step.

Thread two (horizontal): The relationship between diffusion time and physical time. TimeGrad runs the denoising index $n$ inside the $\mathbb{R}^D$ cross-section at each physical time $t$—the two axes are orthogonal. Over the next five years, these axes underwent juxtaposition (CSDI), isomorphic exploitation (MG-TSD), alignment (DYffusion, ARMD), until TEDM in 2026 merged them into one. This geometric relation is the fastest way to locate any time-series diffusion paper.

2. Theoretical Foundation: Five Pillars¶

To read what follows, we prepare five tools. Each gets one formula and one sentence; full derivations, counterexamples, and numerical experiments are treated systematically in [M4AI, Ch. 17–21]—Part IV of that book, "Score Functions, Dynamics, and Diffusion." Chapters 13–16 lay the density perspective through probability, KL divergence, and variational bounds; from Chapter 17 onward we enter the geometric perspective this note needs.

Pillar one: Score function and denoising score matching [M4AI, Ch. 17]. The score function $s(\mathbf{x}) = \nabla_{\mathbf{x}} \log p(\mathbf{x})$ is the local geometry of log-density; it contains no partition function—precisely why energy models are trainable. But real data concentrate near a low-dimensional manifold, where the score is undefined on the manifold itself; the fix is to add noise. For $q_\sigma(\tilde{\mathbf{x}}|\mathbf{x}) = \mathcal{N}(\tilde{\mathbf{x}};\mathbf{x},\sigma^2\mathbf{I})$, the identity

shows that "predict noise $\boldsymbol{\epsilon}$" and "estimate the score" differ only by scaling [4, 5, 6]. Every $\epsilon$-parameterized diffusion model is an implicit score estimator. Multi-scale noise (from large $\sigma$ to small $\sigma$) simultaneously addresses the manifold problem and coverage of low-density regions.

Pillar two: Langevin sampling [M4AI, Ch. 18]. Once we have the score, we must turn it into samples. Unadjusted Langevin algorithm (ULA)

is "gradient ascent + noise injection." Noise cannot be omitted: remove it and the iteration collapses to a mode—an optimizer, not a sampler. The annealed version lets the noise scale decrease from large to small, meshing exactly with the multi-scale structure of pillar one.

Pillar three: Reverse-time SDE [M4AI, Ch. 19]. Treat the discrete noising chain as Euler–Maruyama discretization; its continuous limit is the variance-preserving (VP) SDE $d\mathbf{x} = -\tfrac{1}{2}\beta(t)\mathbf{x}\,dt + \sqrt{\beta(t)}\,d\mathbf{W}$. Anderson (1982) [8] gives its time reversal:

Here the score is not a modeling preference but the price of time reversal: without $\nabla\log p_t$ there is no reverse equation. This pillar explains why pillar one must come first.

Pillar four: Probability-flow ODE and flow matching [M4AI, Ch. 20]. The same family of marginal distributions $\{p_t\}$ can be carried by deterministic dynamics:

DDIM [9] is its discrete counterpart; flow matching [21, 22] goes further, bypassing score and SDE and directly regressing the velocity field connecting $p_0$ and $p_1$—at which point the path and even the reference measure itself become design variables, and optimal transport supplies criteria for choosing paths. Linear state-space models, zero-order hold (ZOH), and bilinear discretization in the same chapter are another appearance of the discrete–continuous duality principle on sequence-model backbones.

Pillar five: Fokker–Planck and three-layer description [M4AI, Ch. 21]. Above the path layer (sample trajectories) and equation layer (SDE/ODE) sits the distribution layer: density $p_t$ obeys the Fokker–Planck equation

Forward diffusion at this layer is an OU process / heat equation; closed-form marginals at any time explain why the terminal state converges to standard Gaussian; the reverse problem is ill-posed at this layer—unless $\nabla\log p_t$ is provided. That is the distribution-layer one-line proof of "why we must learn the score."

The progression among these five pillars is not arbitrary: each presupposes the previous. Below we will see the literature instantiate them in the same order over five years.

3. 2021: Three Initial Conditions¶

3.1 TimeGrad: Establishing Orthogonal Two Axes¶

The entire structure of TimeGrad [1] fits in one factorization:

where $\mathbf{h}_{t-1}$ is an RNN encoding of history and covariates, and each factor is implemented by a conditional DDPM: the denoising network $\boldsymbol{\epsilon}_\theta(\mathbf{x}^n_t, \mathbf{h}_{t-1}, n)$ takes one extra conditional input compared to the image version, and the conditional ELBO derivation carries through unchanged. The training objective is the familiar simplified loss

and the original paper states explicitly that it is "of the same form as the score-matching loss of noise conditional score networks (NCSN)"—precisely the identity of pillar one (systematic derivation in [M4AI, Ch. 17]; the ELBO side in [M4AI, Ch. 14–16]). The sampling algorithm is named annealed Langevin dynamics: inject noise $\mathbf{z}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ when $n>1$, set $\mathbf{z}=\mathbf{0}$ when $n=1$, descending scale stepwise from the noisiest perturbation—a literal engineering implementation of pillar two, including "why noise cannot be omitted" [M4AI, Ch. 18]. Experimentally, TimeGrad set new bests on six public datasets (up to dimension 2000) under CRPS$_{\text{sum}}$; ablations show $N\approx 10$ diffusion steps still usable, $N\approx 100$ optimal—a curve that foreshadowed the entire fast-sampling literature.

An honest assessment: zero new mathematics. Closed-form forward marginals, closed-form posteriors, and $\epsilon$-parameterization all come from Ho et al. [2]; innovation happened at the composition layer—turning DDPM from an unconditional image generator into a general probabilistic emission head for sequence models, sidestepping Gaussian $O(D^2)/O(D^3)$, copula low-rank constraints, and topological limits of flows. But one structural contribution was inconspicuous then and looks like the origin in hindsight: TimeGrad established the orthogonal layout of two time axes—denoising index $n$ runs inside the $\mathbb{R}^D$ cross-section at each physical time $t$, perpendicular to the time direction. Every subsequent time-series diffusion model can be located by how it treats these two axes.

3.2 CSDI: Juxtaposed Two Axes¶

CSDI [10] in the same year offered the first alternative: treat the entire time $\times$ dimension block as one "graph" for unified noising and denoising. Diffusion index $n$ remains independent of $t$, but its scope expands from a single cross-section to the whole block—the axes go from orthogonal to juxtaposed. This choice accomplishes three things at once: forecasting becomes the special case of imputation with "the entire right block missing," unifying imputation and prediction; non-autoregressive generation avoids error accumulation from stepwise rolling; Transformers inside the block can attend along both time and feature directions. The cost is heavier conditioning structure, and block generation abandons the inductive bias of autoregressive extrapolation. From then on, {autoregressive, block} became the first classification axis of this literature.

3.3 ScoreGrad: Discrete to Continuous¶

ScoreGrad [11] takes TimeGrad's discrete chain to its continuous limit: forward written as a conditional SDE, sampling as numerical solution of the reverse-time SDE—a literal implementation of pillar three [M4AI, Ch. 19]; the authors position it as "the first continuous energy-based generative model for time series forecasting." Worth noting: a conditional ODE sampler added later to the codebase achieves ~20 sampling steps and up to ~4.9× speedup without performance loss—among the earliest empirical instances of pillar four (probability-flow ODE [M4AI, Ch. 20]) in time series.

By end of 2021, the four corners of the coordinate system were set: $\{$autoregressive, block$\}\times\{$discrete, continuous$\}$. The next five years are movement and folding on this grid.

4. 2023–2024: Grammar of Guidance Terms¶

In this phase almost no one changed the forward process; all innovation happened in what to add to the reverse drift. The unified grammar comes from linear composability of scores: by Bayes' rule,

conditioning means adding a guidance term to the unconditional score—a joint corollary of pillars one and two ([M4AI, Ch. 17–18]). Papers differ only in where the guidance term comes from:

TimeDiff (Shen & Kwok, ICML 2023) [14] refines conditioning within the non-autoregressive camp, using future mixup and autoregressive initialization to mitigate loss of temporal inductive bias from block generation. TSDiff (NeurIPS 2023) [15] applies composability most purely: train unconditionally, inject conditions at inference via observation self-guidance—conditioning is fully deferred to sampling time, and the model body stays task-agnostic. SSSD (TMLR 2023) [16] replaces the backbone: S4 structured state space carries long-range dependence—readers will recognize the linear state-space models, diagonalization, and ZOH discretization section of [M4AI, Ch. 20] landed in a diffusion context. MG-TSD (ICLR 2024) [18] observes a clean isomorphism: the "fine-to-coarse" degradation direction of forward diffusion is structurally isomorphic to coarse-graining a time series; coarse-grained sequences directly supervise intermediate noise levels—the axes are not merged, but their isomorphism is used as a supervision signal for the first time. TMDM (ICLR 2024) [19] embeds Transformer priors deeply on both sides of conditioning; RATD (NeurIPS 2024) [20] brings retrieval augmentation into the guidance term: embedding retrieval pulls the most relevant reference sequences from a historical bank to guide denoising, targeting limited corpus size, scarce hard cases, and "missing guidance."

One paper from an adjacent field must be recorded on this thread: DYffusion (NeurIPS 2023) [17] couples diffusion steps with temporal interpolation in spatiotemporal field forecasting—forecast and interpolation alternate, and diffusion direction aligns with physical time for the first time. It targets meteorology rather than general benchmarks, so it was long undervalued in the time-series community; in retrospect, it is the earliest signal of merging the two axes.

5. 2025: Awakening of the Reference Measure¶

The question of 2021–2024 was "how to learn the reverse better"; in 2025 the question flipped: what should the forward process be in the first place? In the language of pillar five [M4AI, Ch. 21]: once endpoint distributions are fixed, the path measure in between was never an axiom—it is a design variable.

The vehicle for the turn is flow matching [21, 22] entering time series. TSFlow (ICLR 2025) [23] gives the cleanest argument of the year: standard diffusion uses $\mathcal{N}(\mathbf{0},\mathbf{I})$ as prior, but it sits far from the distribution of time-series data, making transport paths unnecessarily long; replace it with a (conditional) Gaussian process whose kernel matches temporal structure, combine optimal-transport paths with conditional prior sampling, and an unconditionally trained model suffices for probabilistic forecasting. This is a rare mathematical-level refinement in five years—it changes not the network but the reference measure; its backdrop is the "path selection" perspective laid out in [M4AI, Ch. 20] on flow matching and optimal transport. Another implementation of the same idea is NsDiff (ICML 2025) [24]: location-scale noise lets the forward process itself carry a non-stationary prior—the prior stays fixed in form, the noise morphology moves. ARMD (AAAI 2025) [25] is more radical: future sequence as initial state, historical sequence as terminal state, letting sliding evolution itself serve as the diffusion path—diffusion direction is directly interpreted as time direction, a direct prelude to merging the axes. Others such as CGFM (conditional guided flow matching for residual refinement) and D3U (ICLR 2025, decoupling deterministic and uncertain components) [26] occupy engineering niches and are not expanded here.

Once prior, noise morphology, and path all become design variables, only one question remains: lay out the full design space and choose systematically. That is the story of 2026.

6. 2026: Merging the Two Axes and the Design Space¶

TEDM (ICLR 2026) [29] systematically transplants the EDM methodology Karras et al. established in images [27]—decoupling noise schedule, network preconditioning, and sampler into a modular design space optimizable independently—to time series forecasting, and in doing so completes an action left pending for five years: reducing physical time and diffusion time to the same axis (original wording: "reducing the physical time and the diffusion time to the same axis," with theoretical support). Two direct consequences: first, scaling of noise and signal can be computed empirically from data, eliminating external hand-crafted schedules wholesale; second, sampling complexity that was $O(SH)$ ($S$ diffusion steps $\times$ $H$ forecast steps) collapses to linear in the forecast horizon. A lightweight backbone plus low-latency sampling makes this the first scheme on this line that is "deployable in real time" in an engineering sense. Lineage should list DYffusion (2023, spatiotemporal fields) [17], rolling diffusion and ERDM (2025) [28], and ARMD (2025) [25] as precursors: increasing noise along lead time and letting diffusion direction carry time direction are three approximations of the same idea; TEDM turns it into a systematic choice within the design space. The orthogonal two axes TimeGrad erected five years ago fold into one here.

The cost of folding deserves immediate mention: co-axialization means each physical step passes through denoising once in principle—the freedom of "multiple refinements within each step" is surrendered. For strongly multimodal, heavy-tailed cross-sectional distributions, whether this folding is lossless has no distribution-layer answer yet—we list it as an open problem in §8.

SimDiff (AAAI 2026) [30] opens another front: point forecasting. Diffusion models have an awkward fact over five years—advantages in distributional forecasting did not automatically translate to point forecast accuracy, often lagging regression methods. SimDiff's response is subtraction: one Transformer serves as both denoiser and forecaster, fully end-to-end, without external pre-trained or jointly trained regressors; normalization invariance improves adaptation to distribution shift, median-of-means aggregates multiple inferences—turning generative diversity into a resource for mean squared error. The title posture "Simpler yet better" is itself a critique of prior stacked hybrid architectures.

7. Parallel Track and Evaluation Criteria¶

In parallel with all of the above, from 2024 onward time-series foundation models (Chronos [31], Moirai [32], TimesFM [33], etc.) took the industrial forecasting mainstream with the recipe "tokenization + sequence model + large-scale pretraining," and zero-shot capability is their overwhelming selling point. But the form of their probabilistic output must be seen clearly: most use quantile heads or parametric mixture heads to deliver dimension-wise marginals. Marginals are not joint distributions—quantile heads do not deliver a copula. Wherever the problem is inherently about cross-sectional joint structure—risk aggregation, portfolio-level quantiles, tail co-movement, scenario generation—the generative diffusion and flow route still has no substitute. That is its entire relevance to finance, without further rhetoric.

Here we must criticize the scoreboard itself for these five years. Since TimeGrad, the main metric of this literature has been CRPS$_{\text{sum}}$: sum over $D$ dimensions first, then CRPS on the one-dimensional distribution of the sum. It is proper [1, 35]—the true distribution scores optimally in expectation—but proper does not mean sensitive: summing compresses $D$-dimensional dependence into a one-dimensional distribution of the sum; any dependence difference that does not change the distribution of the sum is invisible to it. In other words, much of five years of SOTA competition was built on a metric blunt to the adjective "multivariate." More discerning alternatives already exist: energy score [35] and variogram score [36] are far more sensitive to joint structure. After the axes merge and sampling drops to linear, the cheapest and most respectable next step for this literature may not be a new model but a new scoreboard.

8. Four Open Problems¶

(I) Error compounding under autoregressive rolling. Learned scores carry estimation error, so single-step conditional distributions have quantifiable bias; after autoregressive rolling for $H$ steps, how does bias grow with $H$? Sensitivity of reverse equations to score error has scattered results in unconditional settings, but no quantitative theory under conditional + rolling settings. This is essentially the conditional version of ill-posedness of the reverse-time Fokker–Planck problem [M4AI, Ch. 21]: each step solves a fragile inverse problem and feeds the solution to the next.

(II) Optimality of the reference measure. TSFlow shortens paths via optimal transport, but no proved correspondence exists between "short transport cost" and "good forecast score." Given a forecast loss (CRPS, energy score), what are the optimal prior and path family? A clean variational problem, currently entirely open.

(III) Information cost of axis merging. Under which classes of cross-sectional distributions is TEDM-style co-axialization lossless? For multimodal or heavy-tailed cases, is there a decidable criterion telling us when refinement steps $S>1$ within each step must be retained? Without such a criterion, "linear sampling" on difficult distributions is merely an unaudited discount.

(IV) Discrete–continuous bookkeeping. Training happens at discrete steps, analysis at the continuous limit, deployment returns to discrete—exactly the discrete–continuous duality principle of [M4AI, Ch. 20] in engineering practice. Errors between the three layers (discretization error, score estimation error, rolling compounding error) each have scattered bounds, but no unified ledger. Closing that ledger would give this literature its own numerical analysis.

9. Conclusion: Literature Replays Theory¶

We collect the two threads.

The first is vertical ascent. TimeGrad (2021) works at the algorithm layer—discrete iterations of annealed Langevin; ScoreGrad (2021) rises to the process layer—reverse-time SDE; DDIM-style acceleration and flow matching move the problem to the geometry layer—deterministic flows and transport paths; TSFlow, NsDiff, and TEDM (2025–26) finally choose at the measure and design-space layer. Place this ascent side by side with Chapters 17–21 of [M4AI]—score and energy models, Langevin sampling, stochastic differential equations, probability flow and continuous limits, Fokker–Planck and distribution dynamics—and the correspondence is step by step. This is not the literature copying a textbook; the logic chain had only one way to go: the textbook wrote it in one pass, the field walked it in five years.

The second is horizontal folding. $n$ and $t$: orthogonal (TimeGrad), juxtaposed (CSDI), isomorphic exploitation (MG-TSD), aligned (DYffusion, ARMD), merged (TEDM). One coordinate system holds five years.

Looking ahead, both threads point to the same place. Vertically, what theory has not yet been cashed out in the literature is only quantitative control at the distribution layer—questions (I) and (IV) in §8; horizontally, the axes are merged, and going forward means either opening new axes (multimodal conditioning, cross-frequency, cross-asset-class joint structure) or paying back theoretical debt. Walking the road and writing the road have always been two halves of the same thing.

References¶

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[M4AI] Xuan Xin. Mathematics for AI and Machine Learning. 2026. The book's 21 chapters fall in four parts: linear algebra (Ch. 1–10), differentiation and optimization (Ch. 11–12), probability and information theory (Ch. 13–16, including KL divergence, ELBO, and variational inference), and score functions, dynamics, and diffusion (Ch. 17–21). This note cites Part IV—Chapter 17 (score functions and energy models), Chapter 18 (Langevin dynamics and sampling), Chapter 19 (stochastic differential equations and Itô calculus), Chapter 20 (ODEs, SDEs, and continuous limits of algorithms), Chapter 21 (Fokker–Planck and distribution dynamics)—as [M4AI, Ch. N]. Website: https://math4ai.org/ ; Amazon: https://www.amazon.com/dp/B0GSXVFMLD

Origins and discrete → continuous (2021)
- TimeGrad — Autoregressive Denoising Diffusion Models for Multivariate Probabilistic TS Forecasting, ICML 2021 — https://arxiv.org/abs/2101.12072
- ScoreGrad — Multivariate Probabilistic TS Forecasting with Continuous Energy-based Generative Models — https://arxiv.org/abs/2106.10121
- CSDI — Conditional Score-based Diffusion Models for Probabilistic TS Imputation, NeurIPS 2021 — https://arxiv.org/abs/2107.03502

Conditioning and backbone improvements (2023–24)
- TimeDiff — Non-autoregressive Conditional Diffusion Models for TS Prediction, ICML 2023 — https://arxiv.org/abs/2306.05043
- TSDiff — Predict, Refine, Synthesize: Self-Guiding Diffusion Models, NeurIPS 2023 — https://arxiv.org/abs/2307.11494
- SSSD — Diffusion-based TS Imputation and Forecasting with Structured State Space Models, TMLR 2023 — https://arxiv.org/abs/2208.09399
- MG-TSD — Multi-Granularity TS Diffusion Models with Guided Learning Process, ICLR 2024 — https://arxiv.org/abs/2403.05751

Flow matching era (2025)
- TSFlow — Flow Matching with Gaussian Process Priors, ICLR 2025 — https://arxiv.org/abs/2410.03024
- NsDiff — Non-stationary Diffusion for Probabilistic TS Forecasting, ICML 2025 — https://arxiv.org/abs/2505.04278
- CGFM — Bridging the Last Mile of Prediction: Conditional Guided Flow Matching — https://arxiv.org/abs/2507.07192

Parallel track: foundation models
- Chronos — Learning the Language of Time Series (Amazon) — https://arxiv.org/abs/2403.07815
- Moirai — Unified Training of Universal TS Forecasting Transformers (Salesforce, ICML 2024 Oral) — https://arxiv.org/abs/2402.02592
- TimesFM — A Decoder-only Foundation Model for Time-series Forecasting (Google, ICML 2024) — https://arxiv.org/abs/2310.10688
- Toto (Datadog) has two papers: 2024 technical report https://arxiv.org/abs/2407.07874 ; 2025 open-weights release (Toto-1.0 + BOOM benchmark), 151M parameters, decoder-only, pretraining corpus 4–10× other foundation models — https://arxiv.org/abs/2505.14766

Survey
- Su et al. — Diffusion Models for TS Forecasting: A Survey — https://arxiv.org/abs/2507.14507