{"ID":3050064,"CreatedAt":"2026-06-04T02:13:16.786527022Z","UpdatedAt":"2026-06-06T11:59:53.540122282Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04804","arxiv_id":"2606.04804","title":"The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems","abstract":"Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \\emph{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 \\emph{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 \\textbf{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.","short_abstract":"Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \\emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We sh...","url_abs":"https://arxiv.org/abs/2606.04804","url_pdf":"https://arxiv.org/pdf/2606.04804v1","authors":"[\"Jian Xu\",\"Delu Zeng\",\"John Paisley\",\"Qibin Zhao\"]","published":"2026-06-03T12:30:03Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}