From Sparse Sensors to Full-Field Posteriors: Physics-Prior Bridges for Stochastic PDEs

Reconstructing a stochastic physical field from fixed sparse sensors requires propagating observational constraints into unmeasured regions while preserving posterior uncertainty. We propose a physics-controlled two-stage diffusion-bridge inference framework in which the governing SPDE defines the path prior and sparse observations determine the posterior bridge. We propose a two-stage framework for physics-controlled diffusion-bridge inference. First, a local triangular transport calibrates an all-observation coarse Gaussian reference, yielding a tractable variational path density and unbiased local-window training under a physical reverse-KL objective. Stage 1 provides an important calibration for the coarse Gaussian reference and approximate posterior samples at discretized physical times without backpropagating through full controlled trajectories. Second, a stochastic endpoint network distills the underlying diffusion bridge process controlled by the stochastic PDE based on state pairs within the physics trajectory. At inference, terminal fields are sampled through a short reference prefix followed by a single stochastic endpoint prediction. We derive the conditional training identities and an endpoint error decomposition, establishing a theoretical foundation for connecting local path calibration to efficient terminal-state posterior sampling. Preliminary experiments on stochastic PDE benchmarks, including the one-dimensional Burgers equation, the two-dimensional FitzHugh–Nagumo system, and the Navier–Stokes equations, show promising full-field reconstruction from fixed sparse observations.