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DiffPTS: Rethinking Diffusion ELBO for Probabilistic Time Series Forecasting

Weiwei Ye, Dongyuan Li, Hangchen Liu, Haotong Jiang, Yoshihide Sekimoto, Renhe Jiang

arXiv:2609.32363v2cs.LGcs.AI

TL;DR

Probabilistic time-series forecasting must model complex, time-varying future distributions, but diffusion methods often train adaptive estimators separately from the ELBO. DiffPTS reformulates the ELBO to jointly optimize those estimators and diffusion components; it reports lower CRPS and MSE than prior diffusion baselines across nine benchmarks.

  • Problem

    Diffusion forecasters use adaptive mean and variance estimators for distributional shifts, but often train them with separate losses rather than within a revised ELBO.

    Source: §3.1 · p. 5

  • Method

    DiffPTS reformulates the ELBO under the location-scale noise model, deriving a Gaussian NLL for the estimators and a joint objective for end-to-end training.

    Source: §1 · p. 2, §3.3 · p. 5

  • Results

    Over nine benchmarks, DiffPTS outperforms previous diffusion-based baselines by over 14.53%/16.55% in CRPS/MSE.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The reformulated ELBO provides a unified objective for residual-based and endpoint-based diffusion methods.

    Source: §1 · p. 2

    The work focuses on the optimization objective and does not investigate which network architectures best suit the proposed framework.

    Source: §5 · p. 9

Abstract

from arXiv · show

Probabilistic time series forecasting requires modeling and predicting complex and time-varying distributions. Recently, Denoising Diffusion Probabilistic Model (DDPM)-based approaches have shown promise by equipping the dif- fusion process with pretrained mean and variance estimators to accommodate distributional shift. However, these methods typically follow the standard DDPM framework and consider only partial components of the evidence lower bound (ELBO), treating the training of estimators as designed regression tasks separate from the variational inference framework. To address this, we rethink the ELBO under the Location-Scale Noise Model (LSNM) and find that it naturally induces a Gaussian negative log likelihood objective for the estimators and inherently defines a joint training objective that unifies recent diffusion paradigms for probabilistic forecasting. Building on this principled ELBO reformulation, we propose Diff- PTS, a general framework that enables end-to-end optimization of all components within the ELBO. Across multiple benchmarks, DiffPTS consistently outperforms recent models, achieving state-of-the-art performance with an average CRPS/MSE reduction of over 14.53%/16.55% compared to existing diffusion-based methods. The code is available at https://github.com/wwy155/DiffPTS.

1 Introduction

Diffusion forecasters add adaptive mean and variance estimators for distributional shifts, but their training is often separate from the diffusion ELBO. DiffPTS reformulates the ELBO to jointly optimize its components and reports stronger benchmark performance.

1 Introduction

  • Recent methods train adaptive mean and variance estimators separately or with designed regression targets, leaving their alignment with the variational objective unresolved.The introduction notes NsDiff's future sliding-variance target for its variance estimator.Source: §1 · pp. 1–2
  • On ETTm1, Figure 1 compares prediction intervals and endpoint distributions for TimeGrad, NsDiff, and DiffPTS; the text says NsDiff's sliding-variance target overestimates uncertainty.TimeGrad uses a fixed Gaussian prior, while NsDiff uses an adaptive endpoint.Source: §1 · p. 2, Figure 1 · p. 2
  • DiffPTS reformulates the ELBO under the location-scale noise model to jointly optimize reconstruction, denoising, prior matching, and the estimators.The resulting objective unifies residual-based and endpoint-based methods.Source: §1 · p. 2
  • Over nine benchmarks, DiffPTS outperforms previous diffusion-based baselines by over 14.53%/16.55% in CRPS/MSE.The introduction presents this as evidence for optimizing the full ELBO.Source: §1 · p. 2

2 Background

The background introduces DDPM's ELBO-based denoising process and explains why probabilistic forecasting calls for conditioning on history and representing input-dependent uncertainty.

2.1 Denoising Diffusion Probabilistic Model

  • Standard DDPMs use a fixed Gaussian endpoint, add noise in a forward process, and learn a reverse process that removes it.The ELBO includes reconstruction, prior-matching, and denoising terms; with fixed distributions, the first two can be simplified or ignored in standard training.Source: §2.1 · p. 3
  • DDPM training commonly simplifies the ELBO to a denoising loss, predicting noise sampled through a reparameterized forward process.At inference, samples start from the Gaussian endpoint and pass through the learned reverse process.Source: §2.1 · p. 3

2.2 Probabilistic Time Series Forecasting

  • Probabilistic time-series forecasting models the future trajectory conditional on historical observations, while facing temporal dependence, non-stationarity, and input-dependent uncertainty.Recent approaches use a conditional mean and variance estimator to model residuals or set an adaptive endpoint under a location-scale noise model.Source: §2.2 · pp. 3–4

3 Discussion & Methodology

The methodology argues that adaptive estimators should be governed by a corrected ELBO rather than appended through separate heuristic losses. It derives a shared objective and uses it to train DiffPTS end to end.

3.1 The Missing Puzzle of The ELBO

  • Existing diffusion forecasters add mean or variance estimators but train them with ad hoc supervised losses decoupled from a revised ELBO.The paper frames their theoretical grounding in the variational objective as the missing puzzle.Source: §3.1 · pp. 4–5

3.2 A Unified Formulation of The ELBO

  • The generalized ELBO incorporates learnable mean and variance functions, and under appropriate instantiations covers NsDiff, RDIT, D3U, TMDM, and DDPM.The paper states these methods' ELBOs are structurally equivalent under a shared noise schedule.Source: §3.2 · p. 5

3.3 A General Framework: DiffPTS

  • DiffPTS jointly trains the reverse denoising process and a Gaussian NLL endpoint, whose estimators are trained through the ELBO rather than pretraining or hand-crafted targets.When variance is fixed to one, the Gaussian NLL reduces to the MSE used to train conditional predictors in D3U and TMDM.Source: §3.3 · pp. 5–6
  • DiffPTS training covers reconstruction, denoising, and prior matching; sampling starts from the learned endpoint and iteratively applies reverse transitions.The framework description presents this training procedure as paradigm-agnostic.Source: Figure 2 · p. 6, §3.3 · p. 6

4 Experiments

Experiments compare DiffPTS with diffusion-based forecasting baselines on nine real-world datasets, then examine its sample-level forecasts, ELBO components, and plug-and-play enhancements.

4.1 Experiment Setup

  • Experiments use nine datasets, eight recent baselines, input windows of 168 time steps, and CRPS and MSE to assess uncertainty and point forecasting.ILI uses 32 prediction steps; the other datasets use 192.Source: §4.1 · p. 7

4.2 Main Experiments

  • DiffPTS achieves state-of-the-art performance across nine datasets; on ECL versus NsDiff, CRPS falls 24.8% (0.290 to 0.218) and MSE falls 13.9% (0.209 to 0.180).On Traffic, CRPS falls 28.6% (0.378 to 0.270) and MSE falls 28.4% (0.637 to 0.456).Source: §4.2 · p. 7, Table 2 · p. 8
  • Figure 3’s density comparison uses the final step of the same ETTm1 sample shown in the interval panels.The figure includes historical and estimated densities alongside the empirical distribution and methods’ prediction intervals.Source: Figure 3 · p. 8
  • In the ETTm1 example, DiffPTS’s 96% prediction interval matches the empirical estimate, while several baselines miss history-dependent or step-wise variance patterns.Figure 3 also compares estimated densities with the historical density; DiffPTS aligns more closely than TMDM and D3U, which over- or underestimate variance.Source: §4.2 · p. 8, Figure 3 · p. 8

4.3 Ablation Experiments

  • Across nine datasets, the complete ELBO performs best, while the variant without denoising remains competitive despite modeling only the other terms.The ablations remove mean-estimator NLL, variance-estimator NLL, or denoising loss in turn.Source: §4.3 · p. 9, Table 3 · p. 9
  • Plug-in ELBO training consistently improves NsDiff, RDIT, and D3U across models and datasets; on ETTm1, D3U+DiffPTS reduces CRPS from 0.412 to 0.290 and MSE from 0.589 to 0.357.D3U receives a variance backbone for compatibility with the framework.Source: §4.3 · p. 9, Table 4 · p. 9

5 Conclusion

The conclusion presents DiffPTS as a joint optimization framework derived from the full ELBO under the location-scale noise model. The accompanying derivations establish its component losses and relate its objective to other diffusion paradigms.

5 Conclusion

  • DiffPTS jointly optimizes all ELBO components, whose formulation induces Gaussian negative log-likelihood objectives for adaptive mean and variance estimators.The paper reports consistent improvements over existing approaches and plug-in enhancements that surpass state-of-the-art results.Source: §5 · p. 9

A.1 Proposition 3.1

  • Under the conditional Markov factorization, the ELBO comprises reconstruction, prior-matching, and denoising terms; at finite T, the prior-matching weight remains substantial.With the stated linear schedule and T = 20, the derivation gives ᾱT ≈ 0.82.Source: §A.1 · pp. 14, 16

A.2 Remark 3.2

  • Under a standard denoising schedule, the paper relates the diffusion paradigms of NsDiff, RDIT, D3U, and TMDM through equivalent ELBO objectives.With gψ(X) = 1, RDIT reduces to D3U and NsDiff to TMDM; setting fϕ(X) = 0 further recovers standard DDPM.Source: §A.2 · p. 18

A.3 Proposition 3.3

Proposition 3.3 shows that maximizing the ELBO is equivalent to minimizing a loss combining estimator training and diffusion denoising. Under a data-dependent prior, the convergence analysis gives an optimal Hessian condition number under its stated assumptions.

A.3 Proposition 3.3

  • The ELBO’s reconstruction and prior-matching terms plus denoising terms yield a loss whose first two terms form a Gaussian NLL training fϕ and gψ.The estimator NLL assumes a Gaussian conditional model for Y0 given X.Source: §A.3 · p. 19
  • The expectations in the objective are approximated by sampling a time step and Gaussian noise for each mini-batch.The sampled noise is used to compute Yt.Source: §A.3 · p. 19

A.4 Convergence Rate

  • Under mild constraints and a linear ϵθ, the data-dependent prior achieves Hessian condition number 1 and the fastest possible convergence rate.A simple model suffices for the mean of q(Yt−1|Yt), whereas an isotropic covariance prior requires a more complex model for comparable precision.Source: §A.4 · p. 20

B.1 Datasets Details

The experiments use nine forecasting datasets spanning energy, electricity, traffic, solar, exchange-rate, and influenza-like illness series. They describe estimator implementations and evaluate both point and probabilistic forecasts with standard metrics.

B.1 Datasets Details

  • The benchmark set includes transformer temperature, electricity consumption, exchange rates, traffic, solar output, and influenza-like illness data.The ETT datasets include hourly and 15-minute series; the electricity consumption set covers 321 clients, while traffic data uses 862 sensors.Source: §B.1 · p. 20

B.2 Detail Implementation

  • The mean estimator uses the Non-stationary Transformer, while variance is estimated by a three-layer MLP with a positive-output activation and a 10^-8 variance floor.The MLP hidden dimension is 512; the floor is used for numerical stability.Source: §B.2 · p. 20

B.3 Metrics

  • Evaluation uses MAE and MSE for prediction errors, CRPS to quantify predictive-CDF discrepancy, and QICE to measure calibration by comparing empirical and nominal coverage across quantile intervals.CRPS is estimated using S = 100 samples. QICE partitions the predictive distribution into I non-overlapping, equal-probability quantile intervals, each with nominal coverage 1/I.Source: §B.3 · p. 21

C Additional Showcases

Across nine benchmark datasets, DiffPTS reports consistent state-of-the-art performance, while ablations show degradation when either estimator NLL or diffusion denoising is removed. Its showcase intervals adapt to changing uncertainty patterns across time series.

C Additional Showcases

  • Figure 5’s 96% intervals expand during high variability and contract during stable ETT trends, and widen at sharp EXG directional transitions.The figure shows ETTm2, ETTm1, ETTh2, and EXG samples with ground truth.Source: C Additional Showcases · p. 21, Figure 5 · p. 22

D.1 Main Experiments

  • DiffPTS demonstrates consistent state-of-the-art performance across nine datasets, while NsDiff’s QICE advantage is limited to datasets where its sliding-window variance assumption holds.DiffPTS learns uncertainty patterns directly from data and reports more consistent overall probabilistic forecasting performance.Source: §D.1 · p. 22

D.2 Ablation Results with Error Bar

  • Removing either NLL component or the denoising process degrades performance; removing the mean-estimator NLL causes the largest reported deterioration.On ETTh1, removing the fϕ NLL increases CRPS from 0.374 to 0.553 and MSE from 0.507 to 1.125; the complete model’s error bars are mostly within ±0.01 CRPS and ±0.03 MSE.Source: §D.2 · p. 22

D.3 Full Plug and Play Results

The plug-and-play experiments replace estimator training strategies with the unified ELBO objective in four diffusion forecasters while holding other components and hyperparameters fixed. The resulting variants outperform their respective base models.

D.3 Full Plug and Play Results

  • The objective is tested with NsDiff, RDIT, D3U, and TMDM by replacing estimator training while keeping other components and hyperparameters unchanged.D3U and TMDM also receive the same variance backbone used for DiffPTS.Source: §D.3 · p. 22
  • On D3U, +DiffPTS reduces Traffic CRPS from 0.507 to 0.269 and ILI CRPS from 2.097 to 0.762.The reported relative improvements are 47.0% on Traffic and 63.7% on ILI.Source: §D.3 · p. 24, Table 8 · p. 24

D.4 Efficiency Analysis

DiffPTS reduces training time while adding little memory, and the further analyses clarify the LSNM endpoint assumption. The diagonal Gaussian initializes the generative process; the reverse chain produces the final predictive distribution.

D.4 Efficiency Analysis

  • The efficiency analysis compares training time, memory consumption, and downstream performance for DiffPTS on ETTh1.The experiments otherwise follow Section 4.1 with seeds {1, 2, 3}.Source: §D.4 · p. 24, Table 9 · p. 25, E Further Analyses and Discussions · p. 25
  • An average 55.3% reduction in training time and 15.6% improvement in CRPS come with only 8MB additional memory across TMDM, D3U, and RDIT.The authors attribute the gains to eliminating pretraining through joint optimization.Source: §D.4 · p. 25

E.1 On the LSNM Assumption and the Diagonal Covariance

  • The LSNM defines the conditional endpoint as N(fϕ(X), gψ(X)); the diagonal Gaussian initializes, but does not constrain, the final predictive distribution.The reverse process is conditioned on history and iteratively refines samples, allowing dependencies across time steps and variates; full conditional covariance in the ELBO remains future work.Source: §E.1 · p. 25

E.2 Joint versus Two-Phase Training

The reported comparisons separate the effect of joint ELBO training from architectural differences. Under identical architectures, joint training improves reported forecasting metrics while removing pretraining.

E.2 Joint versus Two-Phase Training

  • DiffPTS inherits its estimator architectures from NsDiff, while D3U contributes a stronger patch-based mean backbone that benefits D3U+DiffPTS on ETT datasets.The reported gap between DiffPTS and D3U+DiffPTS on ETT is architectural; the estimator architectures themselves are not the contribution.Source: §E.2 · p. 25, Table 10 · p. 26
  • On ETTh1, joint ELBO training improves CRPS/MSE from 0.404/0.647 to 0.374/0.507 (7.4%/21.6%), and from 0.331/0.453 to 0.329/0.424 on ETTm1.The comparison holds architecture, noise schedule, and seeds fixed and removes pretraining.Source: §E.2 · p. 26, Table 11 · p. 26
  • D3U uses no variance estimator and implicitly fixes residual variance at one, whereas DiffPTS learns input-dependent variance through the ELBO.DiffPTS trains the mean estimator jointly through the NLL, whose squared error is weighted by 1/gψ(X), and it also receives gradients from the denoising term.Source: §E.2 · p. 26

E.3 Stationary versus Non-Stationary Datasets

The analyses report larger gains from the ELBO objective on non-stationary datasets, compare simplified and exact-weighted objectives, and examine stability, diffusion-step choice, fast sampling, and prediction parameterization.

E.3 Stationary versus Non-Stationary Datasets

  • Gains from +DiffPTS are positive for all four hosts and consistently larger on the non-stationary group than the stationary group.The non-stationary group is Traffic, EXG, ILI, ECL, and ETTm2; the stationary group is ETTh2, ETTm1, ETTh1, and Solar.Source: §E.3 · p. 27, Table 12 · p. 27

E.4 Effect of the ELBO Weighting Coefficients

  • The simplified ELBO-derived objective is equal or better on every dataset than the exact-weighted objective, which applies schedule-determined coefficients to denoising and endpoint NLL terms.The implementation uses the standard simplified weighting, so Equation (14) is not the exact ELBO.Source: §E.4 · p. 27, Table 13 · p. 27

E.5 Numerical Stability of the Gaussian NLL

  • Across nine datasets and three seeds, the authors observed no NaN or divergence, and the learned variance stayed well bounded away from zero on ETTh1 and ETTm1.By contrast, the NsDiff sliding-window variance target can approach zero on locally flat segments, where its Gaussian NLL explodes.Source: §E.5 · pp. 27–28, Table 14 · p. 28

E.6 Choice of the Number of Diffusion Steps

  • T = 20 is best on all datasets; reducing T from 20 to 1 on ETTh1 changes CRPS from 0.377 to 0.387, while removing denoising changes it from 0.374 to 0.437.The authors attribute the modest step requirement to the low-dimensional conditional target and the learned endpoint absorbing location and scale; they conjecture that large T increases estimation burden with scarce, non-stationary data.Source: §E.6 · p. 28, Table 15 · p. 28

E.7 Fast Sampling with DPM-Solver

  • With five DPM-Solver steps, CRPS stays within about 3% of ancestral sampling at roughly 2× speedup.The comparison uses 100 samples, three seeds, and the same checkpoints and evaluation protocol; inference otherwise still iterates the reverse diffusion.Source: §E.7 · p. 29, Table 16 · p. 29

E.8 x0-Prediction versus ϵ-Prediction

  • At T = 20, ϵ-prediction is clearly preferable to x0-prediction, although training remains stable with either parameterization.The paper adopts ϵ-prediction.Source: §E.8 · p. 29, Table 17 · p. 29

E.9 Limitations and Failure Cases

DiffPTS helps less when host estimators are already well specified and is not best on every metric or dataset. The authors also identify modeling and evaluation choices that constrain the approach.

E.9 Limitations and Failure Cases

  • DiffPTS gains are marginal when host estimators are well specified; NsDiff has better QICE on ETTh2 and ILI, while some plug-and-play variants have slightly higher MSE than their hosts on individual datasets.The listed MSE exceptions are TMDM on ETTm2, D3U on EXG, and RDIT on Solar.Source: §E.9 · p. 29
  • The LSNM endpoint uses diagonal covariance, the method follows simplified DDPM weighting, and the number of diffusion steps is chosen empirically.Source: §E.9 · p. 30

NeurIPS Paper Checklist

  • The paper reports that its experiments are reproducible, with setup and implementation details in Section 4 and code publicly available alongside public benchmark datasets.Source: NeurIPS Paper Checklist · pp. 32–33
  • The checklist states that training and test details are discussed in Section 4.1 and Appendix B.2, and results with error bars are provided in Appendix D.2.Source: NeurIPS Paper Checklist · p. 33
  • The authors say hardware details appear in Section 4.1 and an efficiency analysis in Appendix D.4; they also report that dataset creators are credited and repository code is documented.Source: NeurIPS Paper Checklist · pp. 34–36
  • The checklist marks ethics conformity and limitations discussion as affirmed, while societal-impact discussion is marked not applicable because the work focuses on technical contributions.Source: NeurIPS Paper Checklist · pp. 31, 34

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