aiqbench analysis
Current methods that enforce hard physics constraints in generative models for PDE inverse problems introduce a significant statistical bias by omitting a necessary co-area correction factor. This oversight causes posterior error to inflate up to 20 times the sampling-noise floor, proving that satisfying physical equations does not equate to sampling the correct Bayesian posterior. The proposed CoCoS sampler resolves this ambiguity, offering a principled approach for accurate uncertainty-aware scientific inference.
The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
arXiv:2606.04804v1 Announce Type: new 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.
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.
Machine Learning (cs.LG)
arXiv:2606.04804 [cs.LG]
(or arXiv:2606.04804v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.04804 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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