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

arXiv: https://arxiv.org/abs/2606.04804 · HTML: https://mlsanity.com/paper/2606.04804/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 7 of 7)

- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Source-Guided Flow Matching](https://mlsanity.com/paper/2508.14807/) (2025)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (2025)

## Built on by (most cited first, 2 of 2)

- [PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generation](https://mlsanity.com/paper/2609.40287/) (2026)
- [Characterizing and Mitigating Distributional Bias in Physics-Informed Generative Models](https://mlsanity.com/paper/2610.16582/) (2026)

## Abstract (arXiv)

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(JJtop)]-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× the sampling-noise floor; minimal-displacement projection (as in PCFM) is biased at 9× 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.