AI summaryⓘ
The authors address the difficulty of making accurate long-term predictions in systems that change over time and space, where small errors can build up and models often ignore physical rules. They propose PDYffusion, which combines a tool that keeps predictions physically consistent using equations called PDEs with a method (UKF) that carefully tracks uncertainty to avoid error growth. Their theory shows the method smoothly follows the physical laws and that its forecasting improves over time. Tests on various datasets show PDYffusion performs better than existing methods in accuracy and reliable uncertainty estimates. The authors also explain how their method balances the trade-off between being accurate and knowing when to be uncertain.
Long-horizon spatiotemporal predictionDiffusion modelsPartial Differential Equations (PDEs)Unscented Kalman Filter (UKF)Error accumulationUncertainty quantificationCRPS (Continuous Ranked Probability Score)MSE (Mean Squared Error)PDE regularizationSSR (Stable Score Reduction)
Authors
Min Young Baeg, Yoon-Yeong Kim
Abstract
Long-horizon spatiotemporal prediction remains a challenging problem due to cumulative errors, noise amplification, and the lack of physical consistency in existing models. While diffusion models provide a probabilistic framework for modeling uncertainty, conventional approaches often rely on mean squared error objectives and fail to capture the underlying dynamics governed by physical laws. In this work, we propose PDYffusion, a dynamics-informed diffusion framework that integrates PDE-based regularization and uncertainty-aware forecasting for stable long-term prediction. The proposed method consists of two key components: a PDE-regularized interpolator and a UKF-based forecaster. The interpolator incorporates a differential operator to enforce physically consistent intermediate states, while the forecaster leverages the Unscented Kalman Filter to explicitly model uncertainty and mitigate error accumulation during iterative prediction. We provide theoretical analyses showing that the proposed interpolator satisfies PDE-constrained smoothness properties, and that the forecaster converges under the proposed loss formulation. Extensive experiments on multiple dynamical datasets demonstrate that PDYffusion achieves superior performance in terms of CRPS and MSE, while maintaining stable uncertainty behavior measured by SSR. We further analyze the inherent trade-off between prediction accuracy and uncertainty, showing that our method provides a balanced and robust solution for long-horizon forecasting.