The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems

Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a hard constraint (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution.

Conditioning a generative prior on a hard PDE constraint is conditioning on a measure-zero manifold -- an operation that is intrinsically ambiguous (the Borel--Kolmogorov paradox) and whose physically correct resolution, the small-residual-noise limit, carries a co-area (Fixman) Jacobian factor $[det(JJ^{\top})]^{-1/2}$ that projection- and guidance-based methods silently omit. We make the bias precise, show that it grows with the heterogeneity of the constraint sensitivity, and validate it on controlled problems against an i.i.d. ground-truth arbiter. The omitted factor is not a second-order detail: removing it inflates the posterior error to $20\times$ the sampling-noise floor; minimal-displacement projection (as in PCFM) is biased at $9\times$ the floor; and a naive scalar reweighting does not fix it.

We introduce CoCoS, a measure-aware constrained sampler that targets the correct co-area posterior, and show that it matches the gold-standard posterior to within sampling noise. Our results imply that "satisfying the physics" is not the same as "sampling the posterior," and give a principled correction for uncertainty-aware scientific inference.