Differencing the Diffusion Trajectory toward Uncertain Components for Time Series Forecasting

Abstract

Diffusion models have become a widely used framework for probabilistic time series forecasting, modeling the distribution of future values given an observed history. In time series forecasting, however, the future continues the observed history, creating an asymmetry the standard diffusion process leaves unaddressed, with slowly-varying content largely determined by the observed continuity while higher-frequency dynamics carry most of the residual uncertainty. Existing diffusion-based forecasters decouple this asymmetry through an external rule before generation, leaving the corruption trajectory blind to which parts of the target the history can already anchor. We propose DiffDiff, a diffusion framework that embeds this predictability asymmetry into the diffusion trajectory itself, so that a single end-to-end diffusion process becomes aware of which parts of the target the history can already anchor. DiffDiff makes the forward operator step-dependent so that the noisy intermediate state progressively shifts from the target itself toward its second-order differenced structure, while a conditioning pathway supplies the denoiser with both value-domain and differential history information balanced by a stage-adaptive gate at each diffusion step. The terminal distribution approaches a standard Gaussian, preserving compatibility with existing samplers. On seven benchmarks across four prediction horizons, DiffDiff outperforms six diffusion baselines, while our analyses show that its trajectory reallocates modeling effort according to history predictability and improves distributional fidelity.