PDE-regularized Dynamics-informed Diffusion with Uncertainty-aware Filtering for Long-Horizon Dynamics
Summary: arXiv:2604.09058v1 Announce Type: cross
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.
Key Components of PDYffusion
- PDE-regularized Interpolator: This component incorporates a differential operator to enforce physically consistent intermediate states, ensuring that predictions adhere to the underlying physics of the system being modeled.
- UKF-based Forecaster: Leveraging the Unscented Kalman Filter, this forecaster explicitly models uncertainty and aims to mitigate error accumulation during iterative prediction processes.
Theoretical Analysis
We provide theoretical analyses demonstrating that the proposed interpolator satisfies PDE-constrained smoothness properties. Additionally, we show that the forecaster converges under the proposed loss formulation, ensuring reliable long-term predictions.
Experimental Results
Extensive experiments conducted on multiple dynamical datasets reveal that PDYffusion outperforms existing methods in terms of Continuous Ranked Probability Score (CRPS) and Mean Squared Error (MSE). Furthermore, it maintains stable uncertainty behavior as gauged by the Spread-Skill Ratio (SSR).
Trade-off Analysis
Our analysis delves into the inherent trade-off between prediction accuracy and uncertainty. The findings indicate that PDYffusion provides a balanced and robust solution for long-horizon forecasting, effectively addressing the challenges posed by cumulative errors and noise amplification.
Conclusion
In summary, PDYffusion represents a significant advancement in the realm of long-horizon spatiotemporal prediction. By combining PDE-driven regularization with uncertainty-aware forecasting, this innovative approach not only enhances prediction accuracy but also ensures consistency with physical laws. The contributions of this work pave the way for more reliable and interpretable predictive models in various applications, from climate modeling to financial forecasting.