Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervision

Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.

Recovering complete physical fields from sparse observations is challenging because the measurements may not uniquely determine the underlying state. Diffusion-based PDE solvers address this problem through iterative sampling whereas neural operators provide deterministic one-pass predictions. We propose SCOPE (Sparse-Context Observability-aware Predictive Embeddings) to recover complete PDE fields from sparse observations by coupling full-field latent prediction with physical reconstruction. A shared decoder reconstructs fields from both predicted and complete-view representations so that representation learning is guided by both physical recovery and latent matching. We derive a quadratic risk decomposition at fixed teacher-decoder pairs showing why optimal latent prediction need not yield optimal field reconstruction. We also establish sufficient conditions for decoder improvements on complete inputs to transfer to recovery from partial observations. Experiments across five PDE settings show that SCOPE outperforms mask-aware neural operators on all ten forward and inverse tasks and achieves lower errors than those reported for diffusion-based solvers including DiffusionPDE and FunDPS. Decoder-only adaptation further improves recovery without retraining the backbone while retaining deterministic single-pass inference.

Reconstructing physical fields from training samples that are always incomplete requires learning spatial structure from fragmented observations. Existing context--query work establishes how held-out observations provide valid training targets, but this does not make the complete-field distribution identifiable when every training field is incomplete. With finite data, weak evidence of sharp transitions and localized variations can further favor averaged predictions that attenuate local detail. A structural prior is therefore needed to favor plausible completions; local spatial relationships offer one grounded in the observations. We propose a locally constructed, globally revisable estimator that explicitly learns local field estimates and subsequently corrects them using full-domain observations. A shared coordinate-conditioned predictor learns from incomplete patches, allowing relatively well-observed neighborhoods to provide direct supervision of local structure. Its overlapping predictions are reconciled into an observation-conditioned consensus field. A full-domain estimator retains the original observations and learns a residual correction around this frozen field estimate, allowing locally constructed structure to be revised by broader evidence. The local estimate serves as both an explicit input, accompanied by its discrepancies with the observations, and a prediction starting point that the global model can revise. On three real-world ocean datasets with authentic observation gaps, our estimator achieves the lowest MSE and highest PSNR on withheld source-supported values, reducing MSE by 28.9%--34.5% against the strongest external baseline.