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GARDiff: Graph-Aligned Residual Diffusion for Probabilistic Multivariate Time-Series Forecasting

Rui Han, Min Yang, Xu Zhang, Xinghao Yang, Wei Liu, Yongshun Gong

arXiv:2609.37694v1cs.LGcs.AI

TL;DR

Directly transferring deterministic-derived graphs to residual generation is unreliable because their edge structure can misalign with residual dependencies. GARDiff progressively refines those graphs using residual uncertainty and diffusion timestep; across six benchmarks, it outperforms strong forecasting baselines on probabilistic metrics.

Abstract

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Diffusion models have recently shown strong potential for probabilistic multivariate time-series forecasting by modeling complex conditional distributions. Recent decoupled diffusion frameworks further separate forecasting into deterministic prediction and stochastic residual generation, making it natural to derive dependency graphs from deterministic representations and use them to guide residual diffusion. However, we show that this direct structural transfer is unreliable. Although deterministic-derived graphs encode useful global dependency priors, they exhibit substantial edge-level misalignment with residual dependency structures, introducing inaccurate or redundant conditions during residual generation. This reveals a previously overlooked deterministic-to-residual structural alignment problem in decoupled diffusion forecasting. To address this problem, we propose GARDiff, a Graph-Aligned Residual Diffusion framework for probabilistic multivariate time-series forecasting. Instead of treating deterministic-derived graphs as fixed diffusion conditions, GARDiff progressively adapts them to residual generation. Specifically, GARDiff estimates residual uncertainty to distinguish high- and low-uncertainty regions, enabling uncertainty-aware structural refinement, and further performs timestep-aware edge sparsification during reverse diffusion to evolve graph conditions from broad dependency aggregation to localized residual refinement. Extensive experiments on six real-world benchmarks demonstrate that GARDiff consistently improves probabilistic forecasting performance and uncertainty calibration over strong baselines.

Graph-Conditioned Diffusion Models

Graph-conditioned diffusion methods use graph topology to guide structured dependency modeling, but fixed graph conditions can miss the changing needs of residual denoising. GARDiff derives graphs from deterministic representations and refines them for residual diffusion.

Graph-Conditioned Diffusion Models

  • Existing graph-conditioned methods use topology to guide denoising, but commonly depend on predefined physical graphs unavailable in many general multivariate time-series settings.Examples include DiffSTG, PriSTI, and SpecSTG, which use graph priors when reliable topology is available.Source: Graph-Conditioned Diffusion Models · pp. 2–3
  • GARDiff derives graphs from deterministic forecasting representations and refines them through uncertainty-aware region selection and timestep-aware edge sparsification.Unlike methods relying on predefined physical graphs, this approach adapts structural guidance during reverse diffusion.Source: Graph-Conditioned Diffusion Models · p. 3
  • GARDiff separates forecasting into a deterministic prediction and residual-generation stage, then uses deterministic-stage structural information to guide residual diffusion.The deterministic prediction captures predictable temporal patterns, while diffusion generates stochastic residuals.Source: Overview · p. 3
  • The deterministic dependency graph is built from intermediate forecasting representations, but may contain dependencies irrelevant to residual generation if reused unchanged.GARDiff instead treats it as a global structural prior to align for residual generation.Source: Overview · p. 3
  • The aligned graph conditions residual denoising, with patch-level residual uncertainty and reverse diffusion timestep identified as alignment inputs.The deterministic forecasting representations also serve as conditional context for the denoising model.Source: Overview · p. 3

Deterministic Forecasting and Graph Construction

GARDiff uses intermediate deterministic forecasting representations to extract a global inter-variable dependency backbone. It constructs a sparse, gated graph that supports point forecasting while motivating later alignment for residual generation.

Deterministic Forecasting and Graph Construction

  • Intermediate deterministic representations capture a global backbone of inter-variable dependencies and support both point forecasting and structural guidance for residual diffusion.The graph is learned from representations rather than treating deterministic forecasting solely as a prediction module.Source: Deterministic Forecasting and Graph… · p. 3
  • The method divides each variable’s input series into non-overlapping patches of length p and projects them into a latent space.Patch-level representations provide units for dependency modeling and deterministic forecasting.Source: Deterministic Forecasting and Graph… · p. 3
  • A global context anchor calibrates patch representations before pairwise affinities are computed and dominant dependencies are retained by Top-K sparsification.The context anchor is the mean feature over all patches.Source: Deterministic Forecasting and Graph… · p. 3
  • A structure-gated mechanism adaptively reweights candidate dependencies to produce the deterministic-derived graph, which then enhances deterministic forecasting representations.The graph uses a dynamic structural mask.Source: Deterministic Forecasting and Graph… · p. 3
  • Directly using the deterministic graph for residual diffusion can introduce structural bias because it is optimized for deterministic prediction rather than residual generation.The graph captures stable global dependencies, but may not match residual-generation needs.Source: Deterministic Forecasting and Graph… · p. 3

Unified Graph Alignment Module

GARDiff retains the deterministic graph as a useful global prior but refines it for residual generation, using residual uncertainty and diffusion timestep to guide edge selection. The unified alignment provides structural conditioning that adapts to residual dependencies and the evolving reverse-diffusion objectives.

Unified Graph Alignment Module

  • GARDiff treats the deterministic graph as a constrained refinement prior, avoiding both loss of useful structure and reuse of residual-incompatible dependencies.The prior is neither fully reliable nor entirely invalid.Source: Unified Graph Alignment Module · p. 3
  • Patch-level residual magnitude provides an uncertainty proxy; at inference, a lightweight network need only estimate relative magnitude, not the full residual distribution.True residuals are available during training, while inference uses the proxy network.Source: Unified Graph Alignment Module · pp. 3–4
  • High-uncertainty patches preserve candidate correction paths, while low-uncertainty regions can suppress redundant connections under a dynamic threshold.The threshold is the uncertainty mean plus a tunable multiple of its standard deviation.Source: Unified Graph Alignment Module · p. 4
  • Top-K sparsity changes across reverse diffusion: early noisy stages retain higher connectivity for coarse global structure, while later stages favor localized refinement.The schedule begins at Kmax and reaches Kmin at convergence, with maximum connectivity during the specified early-stage regime.Source: Unified Graph Alignment Module · pp. 4–5
  • The uncertainty-preserving and timestep-sparse masks are combined so edges selected by either are retained, producing adaptive structural conditioning for residual diffusion.The resulting graph is described as aligned with both residual dependency patterns and the evolving reverse-diffusion objectives.Source: Unified Graph Alignment Module · p. 5

Graph-Conditioned Residual Diffusion

GARDiff conditions residual diffusion on a graph that is progressively aligned across reverse-diffusion steps. Graph-gated attention refines noisy residual features before PatchDN denoises them, and sampled residuals are added to the deterministic forecast.

Graph-Conditioned Denoising

  • The aligned graph G(t) updates at each reverse-diffusion step, moving from dense global interactions early to sparse local refinement as diffusion converges.This provides timestep-aware structural conditions to the conditional denoising process.Source: Graph-Conditioned Residual Diffusion · p. 5

Graph-Guided Feature Refinement

  • Graph-gated multi-head attention uses G(t) to modulate self-attention scores, then adds refined features back to noisy residuals through a residual connection.Noisy residual patches are first projected into latent representations with timestep embeddings.Source: Graph-Conditioned Residual Diffusion · p. 5

PatchDN Backbone

  • PatchDN processes graph-enhanced features to predict injected noise or the clean residual, depending on the diffusion parameterization.After reverse diffusion, the denoised residual is added to the deterministic forecast.Source: Graph-Conditioned Residual Diffusion · p. 5

Sampling and Optimization

  • At inference, multiple residual samples are added to the deterministic prediction to approximate the predictive distribution.The model is trained with a joint objective for residual denoising, deterministic forecasting, uncertainty estimation, and graph regularization.Source: Graph-Conditioned Residual Diffusion · p. 5

Experimental Setup

GARDiff is evaluated on six multivariate time-series datasets using probabilistic and deterministic forecasting metrics. The setup compares representative deterministic and probabilistic baselines and uses 100 diffusion samples at inference.

Datasets

  • The six datasets are ETTm1, ETTm2, Weather, Solar-Energy, Electricity, and Traffic, spanning diverse temporal dynamics.The paper identifies strong seasonality and complex inter-variable dependencies among the covered dynamics.Source: Experimental Setup · p. 6

Baselines

  • Baselines include five deterministic methods and four probabilistic methods, including D3U.The deterministic set includes PatchTST, TimesNet, iTransformer, TimeBridge, and TimeFilter; probabilistic methods include TimeDiff, TMDM, NsDiff, and D3U.Source: Experimental Setup · p. 6

Evaluation Metrics

  • Probabilistic forecasting is assessed with CRPS and CRPSsum, while deterministic forecasting uses MSE and MAE; lower values are better.Source: Experimental Setup · p. 6

Implementation Details

  • Inference approximates predictive distributions with 100 diffusion samples, using T = 100 diffusion steps and a linear noise schedule.The uncertainty-aware alignment uses α = 2.0 and continuous Top-K scheduling with Kmin = 4, Kmax = 32, and η = 0.75.Source: Experimental Setup · p. 6

Evaluation Scope

  • The oracle-residual comparison uses input length H=96 and averages results over horizons L ∈{96, 192, 336}.Source: Table 3 · p. 7

Overall Results

GARDiff improves probabilistic forecasting across all six benchmarks and reports competitive deterministic performance across most benchmarks. The reported probabilistic gains are relative to the strongest diffusion baseline, D3U.

Probabilistic Forecasting Performance (RQ1)

  • Up to 5.5% lower CRPS and 41.6% lower CRPSsum than D3U, with GARDiff outperforming all competing methods across six benchmarks.The authors state that these gains suggest improved distributional calibration across diverse temporal patterns.Source: Overall Results · p. 7

Deterministic Forecasting Performance (RQ2)

  • GARDiff achieves competitive deterministic performance across most benchmarks, with consistent MSE and MAE improvements over TimeFilter and PatchTST.The reported strengths are particularly on ETTm1, ETTm2, and Solar.Source: Overall Results · p. 7

Ablation Study (RQ3)

Ablations on ETTm1 and Weather show degradation when graph conditioning, graph alignment, or timestep-aware pruning is removed. The variants test graph removal, fixed deterministic graphs, and constant sparsity.

Ablation Variants

  • The fixed-pruning variant applies constant top-8 sparsity across denoising steps, while the no-alignment variant directly uses the deterministic graph as a fixed condition.Source: Ablation Study (RQ3) · p. 7

Ablation Study (RQ3)

  • All three ablated variants degrade performance on ETTm1 and Weather, supporting the contributions of graph guidance, graph alignment, and adaptive pruning.Removing graph conditioning impairs inter-variable dependency capture; using an unrefined deterministic graph yields suboptimal topology, while static sparsity does not adapt to denoising objectives.Source: Ablation Study (RQ3) · p. 7

Uncertainty Quality Analysis (RQ4)

The uncertainty estimator closely matches oracle residual classifications and serves as a reliable proxy for downstream forecasting.

Uncertainty Quality Analysis (RQ4)

  • EMR is 0.92 on ETTm1 and 0.94 on Weather, measuring patches classified into matching high- or low-uncertainty groups.The reported results indicate that estimated uncertainty captures local forecasting difficulty.Source: Uncertainty Quality Analysis (RQ4) · p. 7
  • Predicted uncertainty and oracle residuals yield highly consistent results across all metrics on both datasets.The comparison is reported in Table 3.Source: Uncertainty Quality Analysis (RQ4) · p. 7
  • On 96-step ETTm1, CRPS is 0.254 with predicted uncertainty versus 0.250 with oracle residuals.The paper concludes that predicted uncertainty acts as a reliable oracle-residual surrogate without degrading performance.Source: Uncertainty Quality Analysis (RQ4) · p. 7

Parameter Sensitivity Analysis (RQ5)

GARDiff is robust across tested values of α and Kmin, with best performance at moderate settings; the paper reports improvements over strong baselines on six benchmarks.

Parameter Sensitivity Analysis (RQ5)

  • Performance is stable across a wide range of α and Kmin, with best results at moderate values.Both parameters control uncertainty-aware graph construction and sparsity.Source: Parameter Sensitivity Analysis (RQ5) · p. 7
  • Too-small or too-large α and Kmin degrade performance through over-dense or over-sparse graph structures.Source: Parameter Sensitivity Analysis (RQ5) · p. 7
  • GARDiff consistently improves probabilistic forecasting performance and uncertainty calibration over strong baselines on six real-world benchmarks.This is the paper’s reported overall result.Source: Conclusion · p. 8

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