Quantile-Guided Diffusion for Probabilistic Time-Series Forecasting

Recent probabilistic weather forecasters train stochastic predictors with the continuous ranked probability score (CRPS) to generate each ensemble member in a single forward pass. These models learn the predictive distribution from the forecast context alone, which becomes difficult at longer forecast horizons where uncertainty is high. To learn the predictive distribution more effectively, we introduce auxiliary conditional denoising tasks that predict the same future state from the context and its corrupted version, which provides partial future information that can reduce prediction ambiguity. Building on distributional diffusion models, we learn the conditional distributions of these tasks with a single stochastic predictor by minimizing a proper scoring rule across noise levels. At inference, the predictor can still generate each ensemble member in a single forward pass at the fully corrupted endpoint. Standard CRPS training is recovered as the endpoint-only special case of our formulation, so our framework extends existing CRPS-based forecasters with only additional conditioning inputs. Controlled experiments show that the auxiliary tasks improve one-step forecasting across architectures, with larger gains at longer forecast horizons. The gains extend to high-dimensional global weather forecasting under both training from scratch and fine-tuning, along with improved calibration and potential benefits for generalization under distribution shift.

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.

Diffusion models offer a natural way to model uncertainty in time series forecasting, yet their iterative sampling process is often treated as a uniformly beneficial refinement procedure. Our study challenges this view by examining how forecast quality evolves throughout reverse diffusion. We find that general temporal structure is often recovered at relatively high noise levels, whereas continued low-noise refinement can introduce statistical drift and degrade the final forecast. Our analysis further suggests that this behavior explains why prior methods often favor relatively narrow diffusion architecture and schedule design. Building on this observation, we propose a label-free global stopping criterion that detects the optimal termination point, eventually speeding up inference and improving predictive accuracy. Additionally, since early stopping terminates inference in high-noise regions, we propose a Bernoulli timestep sampler that concentrates training on this region while preserving coverage of the full diffusion process. Extensive experiments conducted across eight real-world datasets demonstrate the superior performance of our method compared to existing approaches.