Learning Spatiotemporal Diffusion Models with Continuous-time PDE Guidance

Spatiotemporal Partial Differential Equations (PDEs) are fundamental to modeling complex physical systems. While significant progress has been made in discovering these dynamics from observed data (the ''discovery task''), reconstructing a trajectory of entire fields from sparse, arbitrary observations—the ''imputation task''—remains a substantial challenge. We propose a novel framework that first learns the unknown PDE dynamics using a data-driven approach, specifically by adapting the SINDy method. Subsequently, we employ a generative denoising diffusion model to impute complete spatiotemporal fields. Dual-guidance of the generative process by sparse observations and learned PDE dynamics enables accurate and physically consistent reconstruction of spatiotemporal phenomena even when both the dynamics and the initial conditions are unknown.