The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
2026-06-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explain that common generative models used to solve PDE inverse problems by strictly enforcing physics constraints actually produce biased results. This happens because conditioning on exact PDE constraints involves tricky math related to zero-measure sets, and standard methods miss an important correction factor (called the Fixman Jacobian) that affects the distribution. They demonstrate that ignoring this factor leads to large errors in uncertainty estimates. To fix this, the authors propose CoCoS, a new sampler that includes the correct mathematical adjustment and produces accurate, unbiased posterior samples.
Generative modelsPartial Differential Equations (PDEs)Inverse problemsBayesian posteriorMeasure-zero manifoldBorel–Kolmogorov paradoxCo-area formulaFixman JacobianUncertainty quantificationConstrained sampling
Authors
Jian Xu, Delu Zeng, John Paisley, Qibin Zhao
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.