APOD: Adaptive PDE-Observation Diffusion for Physics-Constrained Sampling

In many parametric PDE problems, partial or noisy observations pose a serious challenge for building robust world models that also respect physical constraints. We introduce Adaptive PDE-Observation Diffusion (APOD), a novel framework that dynamically couples PDE constraints with measurement data during the reverse diffusion sampling process. At each denoising iteration, APOD balances a PDE-consistency term derived from governing equations and a data-fidelity term informed by partial observations, guiding the model to produce physically valid solutions. This balanced enforcement naturally handles sparse or noisy data, alleviates mismatches between PDE residuals and diffusion steps, and enhances solution diversity. Empirical results demonstrate APOD's ability to yield accurate, reliable solution even under uncertainty and limited measurements. Our approach paves a principled way to generate high-fidelity parametric PDE solutions in world-model-based reasoning for scientific and engineering domains.