Numerical Calibration for Physics-Informed Diffusion Posterior Sampling

Physics-informed diffusion posterior sampling aims to improve the physical consistency of reconstructions by incorporating physical constraints into the reverse process of a pretrained diffusion model. These constraints are imposed through physics guidance, which acts alongside the prior score during posterior sampling to steer the reverse process. However, physics guidance requires an explicit numerical realization of the PDE at inference time, whereas the numerical realization underlying the pretrained prior is typically overlooked when constructing the guidance and may even be entirely unknown. We show that this numerical asymmetry matters at finite resolution: different numerical realizations of the same continuous PDE can produce substantially different residuals and physics gradients for the same solution estimate, especially near the resolution limit. The resulting numerical mismatch can cause physics guidance to conflict with the prior score and observation guidance, ultimately degrading posterior reconstruction. These findings suggest a simple principle: when the numerical realization underlying the prior is known, physics guidance should use a matched realization. When it is unknown, we introduce Numerically Calibrated Physics Guidance (NCPG), which learns the numerical relation implicit in paired training data without access to the original solver or knowledge of its discretization. Once learned, the calibration model is frozen and reused across posterior sampling tasks. Experiments on linear and nonlinear PDEs show that numerical mismatch can substantially degrade posterior sampling performance, while NCPG largely mitigates this degradation and, in some settings, even outperforms numerically matched physics guidance.