Researchers Identify and Correct Fundamental Bias in Physics-Constrained Generative Models for PDE Inverse Problems

A preprint posted to arXiv argues that a standard recipe in scientific machine learning — using generative models such as diffusion models or flow matching to solve PDE inverse problems with hard physics constraints — is fundamentally miscalibrated. When a physical law is enforced as an exact constraint, the conditioning operation occurs on a measure-zero manifold, which is mathematically ambiguous without additional specification. The physically correct resolution requires including a co-area Jacobian factor, proportional to [det(JJ⊤)]^{-1/2}, that existing projection- and guidance-based methods silently omit. The authors quantify the resulting bias, showing it scales with the heterogeneity of constraint sensitivity: uncorrected projection inflates posterior error to 20 times the sampling-noise floor, while minimal-displacement projection (as used in PCFM) still errs at 9 times the floor, and naive scalar reweighting fails to resolve the problem. To address this, the authors introduce CoCoS (Co-area Corrected Sampler), a measure-aware constrained sampler that incorporates the missing Jacobian and is shown to match a gold-standard i.i.d. posterior to within sampling noise on controlled test problems. The work draws a sharp distinction between 'satisfying the physics' and 'correctly sampling the Bayesian posterior,' with direct implications for any scientific application relying on these methods for uncertainty-aware inference.