[2604.09058] PDE-regularized Dynamics-informed Diffusion with Uncertainty-aware Filtering for Long-Horizon Dynamics

Computer Science > Machine Learning arXiv:2604.09058 (cs) [Submitted on 10 Apr 2026] Title:PDE-regularized Dynamics-informed Diffusion with Uncertainty-aware Filtering for Long-Horizon Dynamics Authors:Min Young Baeg, Yoon-Yeong Kim View a PDF of the paper titled PDE-regularized Dynamics-informed Diffusion with Uncertainty-aware Filtering for Long-Horizon Dynamics, by Min Young Baeg and Yoon-Yeong Kim View PDF HTML (experimental) 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 smooth...