PDE-regularized Dynamics-informed Diffusion with Uncertainty-aware Filtering for Long-Horizon Dynamics

2026-04-10

AI Summary

This work addresses the challenges of cumulative errors, noise amplification, and physical inconsistency in long-term spatiotemporal forecasting by proposing PDYffusion, a novel framework that integrates a partial differential equation (PDE)-regularized interpolator and an unscented Kalman filter (UKF)-based forecaster into a diffusion model. By explicitly modeling uncertainty and incorporating physical priors, PDYffusion theoretically guarantees that interpolated states adhere to physical laws and that forecasts converge. The method further reveals an intrinsic trade-off between prediction accuracy and uncertainty quantification. Empirical evaluations demonstrate that PDYffusion significantly outperforms existing approaches across multiple dynamical systems datasets, achieving superior performance in both Continuous Ranked Probability Score (CRPS) and Mean Squared Error (MSE), while maintaining well-calibrated uncertainty estimates.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Relational Probabilistic ModelsPlanning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: Federated recommendation systems and personalizationSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• long-horizon prediction
• spatiotemporal dynamics
• physical consistency
• uncertainty quantification
• cumulative errors

Innovation

• PDE-regularization
• diffusion model
• Uncertainty-aware forecasting
• Unscented Kalman Filter
• long-horizon prediction