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

2026-06-03

AI Summary

This work addresses a systematic bias in existing generative-model-based approaches to solving inverse problems for partial differential equations (PDEs), which arises when sampling the posterior under hard physical constraints while neglecting the Fixman (coarea) Jacobian factor induced by conditioning on measure-zero manifolds. The study is the first to explicitly quantify this measure-theoretic bias and introduces CoCoS, a measure-aware sampler that incorporates an explicit coarea correction. By integrating hard PDE constraints with asymptotic analysis in the small-residual limit within diffusion and flow-matching frameworks, CoCoS rigorously recovers the correct Bayesian posterior distribution. Experiments demonstrate that omitting this Jacobian factor leads to posterior errors up to 20 times the sampling noise floor, whereas CoCoS reduces errors to near-noise levels, substantially outperforming current methods.

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.

Problem

• PDE inverse problems
• generative models
• posterior consistency
• co-area correction
• measure-zero constraints

Innovation

• co-area correction
• PDE inverse problems
• generative modeling
• posterior consistency
• measure-theoretic sampling