Learning Continuous-time Dynamics Guidance for Diffusion-based Spatiotemporal Reconstruction

We address the ill-posed inverse problem of reconstructing complete spatiotemporal fields from sparse measurements in systems governed by unknown PDEs. Diffusion models have emerged as powerful generative models for such inverse problems. However, conditioning on sparse measurements alone can produce reconstructions that match the observations but remain inconsistent with the underlying dynamics, limiting their reliability in unobserved regions of the domain. We propose DiffSINDy, a guided diffusion posterior sampling framework in which a diffusion model provides a prior over full spatiotemporal trajectories, while a learned surrogate provides dynamics guidance without requiring the ground-truth governing equations. We instantiate the surrogate with Sparse Identification of Nonlinear Dynamics (SINDy) and couple it with a differentiable method-of-lines solver, yielding an interpretable continuous-time guidance model that can be integrated between consecutive snapshots even when the data are temporally sparse. We evaluate the method on ODE and PDE benchmarks against guidance based on neural surrogates, two oracle variants with true PDEs, and observation-only DPS. Across benchmarks, SINDy-guided diffusion achieves competitive reconstruction error and improves physical consistency over learned neural surrogates and observation-only guidance.