Dynamical Diffusion: Learning Temporal Dynamics with Diffusion Models

Diffusion models have emerged as powerful generative frameworks by progressively adding noise to data through a forward process and then reversing this process to generate realistic samples. While these models have achieved strong performance across various tasks and modalities, their application to temporal predictive learning remains underexplored. Existing approaches treat predictive learning as a conditional generation problem, but often fail to fully exploit the temporal dynamics inherent in the data, leading to challenges in generating temporally coherent sequences. To address this, we introduce Dynamical Diffusion (DyDiff), a theoretically sound framework that incorporates temporally aware forward and reverse processes. Dynamical Diffusion explicitly models temporal transitions at each diffusion step, establishing dependencies on preceding states to better capture temporal dynamics. Through the reparameterization trick, Dynamical Diffusion achieves efficient training and inference similar to any standard diffusion model. Extensive experiments across scientific spatiotemporal forecasting, video prediction, and time series forecasting demonstrate that Dynamical Diffusion consistently improves performance in temporal predictive tasks, filling a crucial gap in existing methodologies. Code is available at this repository: https://github.com/thuml/dynamical-diffusion . * Equal contribution. METHOD We observe that, when integrating diffusion models into predictive learning, there are two notable axes along which the model must learn simultaneously. The first axis, referred to as the "prediction axis", requires the model to learn the temporal dynamics of the data. The second axis, termed the "denoising axis", necessitates that the model distinguishes noise from corrupted states. From this perspective, we identify a mismatch in previous methodologies. As shown in Figure 1a , the forward process in standard diffusion models progresses solely along the denoising axis. In particular, historical observations x -P :0 0 serve only as conditions for denoising networks, with no