cs.LGApr 10, 2026

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

Authors: Min Young Baeg, Yoon-Yeong Kim

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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.

Figures & tables

Figure 1: PDYffusion Framework. Given an intial state xtx_{t} , the interpolator generates intermediate states by enforcing physical structure through the operator (E−l−2Δ)α/2\bigl(E-l^{-2}\Delta\bigr)^{\alpha/2} . These intermediate states are then used by the forecaster to predict the future state xt+hx_{t+h} , while a UKF-based prediction process models and corrects uncertainty via sigma points.
Table 1: Quantitative results on benchmark datasets
MethodSSTNavier–StokesSpring-meshWave
CRPSMSESSRCRPSMSESSRCRPSMSESSRCRPSMSESSR
Dirichlet0.2640.1810.6710.059\bm{0.059}0.017\bm{0.017}0.7310.0092\bm{0.0092}4.01e−4\bm{4.01\mathrm{e}{-4}}1.204\bm{1.204}1.94e−31.94\mathrm{e}{-3}2.33e−52.33\mathrm{e}{-5}0.853
Neumann0.2320.1720.6650.0630.0200.735\bm{0.735}0.01054.16e−44.16\mathrm{e}{-4}1.2112.05e−32.05\mathrm{e}{-3}2.18e−52.18\mathrm{e}{-5}1.110
Periodic0.226\bm{0.226}0.166\bm{0.166}0.683\bm{0.683}0.1070.0330.7120.01214.39e−44.39\mathrm{e}{-4}0.7761.82e−3\bm{1.82\mathrm{e}{-3}}1.57e−5\bm{1.57\mathrm{e}{-5}}0.913\bm{0.913}
Table 2: Experiment results on boundary conditions
λ\lambdaCRPSMSESSR
λ=0.1\lambda=0.10.2410.2410.1780.1780.6100.610
λ=0.2\lambda=0.20.2260.2260.1660.1660.6830.683
λ=0.5\lambda=0.50.3910.3910.3010.3010.5220.522
λ=1.0\lambda=1.00.4380.4380.3260.3260.5240.524
Table 3: Ablation of λ\lambda on SST
Figure 2: Trade-off between accuracy and stability on SST dataset
Figure 3: Wave trajectory results of PDYffusion and DYffusion
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

SymbolUsed for
EEIdentity Matrix
uuGroundtruth solution of PDE
Δ\DeltaLaplacian
ttIndex of temporal data snapshots
hhTraining horizon
NNTotal number of diffusion steps
Appendix
Table 4: Notation for PDYffusion
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