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

• URL: http://arxiv.org/abs/2606.04804v3
• Date: Sun, 07 Jun 2026 13:30:14 GMT
• Title: The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
• Abstract summary: Generative models are increasingly used to solve partial differential equation (PDE) inverse problems.<n>We show that this widely adopted recipe samples the wrong distribution.<n>We introduce textbfCoCoS, a measure-aware constrained sampler that targets the correct co-area posterior.
• Score: 28.97171921966014
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• What Do Flow-Based Inverse Solvers Approximate? A Posterior-Transport View [31.082191748525137]
  We give a posterior-transport flow-based problem solving account.<n>We show that trajectory-guidance solvers can be read as the minimum-energy emphcorrection field needed to morph the unconditional source into the posterior.
  arXiv  Detail & Related papers  (2026-06-23T12:50:11Z)
• Constrained Diffusion Models with Primal-Dual Inference [76.64466170898369]
  We develop constrained diffusion models with primal-dual inference (PDI)<n>PDI infers the optimal primal distribution and its parametrizing dual variable.<n>We evaluate PDI on constrained sampling from a mixture of Gaussians, wireless resource allocation, and portfolio management.
  arXiv  Detail & Related papers  (2026-06-15T18:32:33Z)
• Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory [2.148615003900331]
  We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients.<n>Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.
  arXiv  Detail & Related papers  (2026-05-25T07:14:11Z)
• Bandit Convex Optimization with Gradient Prediction Adaptivity [56.816177049016794]
  We investigate whether optimistic gradient predictions can improve worst-case regret guarantees in a prediction-adaptive manner.<n>The key idea is a novel variance-reduced gradient estimator whose variance scales with the prediction error rather than the gradient norm.<n>We establish an information-theoretic lower bound that scales as $(sqrtmathbbE[S_T])$, providing a fundamental characterization of the best achievable prediction-adaptive regret.
  arXiv  Detail & Related papers  (2026-05-21T08:57:38Z)
• Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability [13.640937605089865]
  Diffusion-based posterior samplers use pretrained diffusion priors to sample from measurement- or reward-conditioned posteriors.<n>This paper introduces a tractable surrogate path connecting the true posterior to a standard Gaussian and comparing it to the sampler's path.<n>A Feynman-Kac representation then expresses the Radon-Nikodym correction as an explicit path expectation, identifying which posterior regions are over- or under-sampled.
  arXiv  Detail & Related papers  (2026-05-07T16:37:29Z)
• Conditional Diffusion Under Linear Constraints: Langevin Mixing and Information-Theoretic Guarantees [10.73598638822051]
  We study zero-shot conditional sampling with pretrained diffusion models for linear inverse problems.<n>We prove that the error from replacing this score by the unconditional tangent score is upper bounded by a dimension-free conditional mutual information between observed and unobserved components.
  arXiv  Detail & Related papers  (2026-05-06T19:19:54Z)
• Dual-space posterior sampling for Bayesian inference in constrained inverse problems [5.739708455652395]
  Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy and incomplete data or inherent non-uniqueness.<n>We develop a new technique for translating hard physical constraints, such as the wave equation, into prior distributions amenable to existing sampling techniques.
  arXiv  Detail & Related papers  (2026-02-28T00:34:31Z)
• Plug-and-Play Diffusion Meets ADMM: Dual-Variable Coupling for Robust Medical Image Reconstruction [45.25461515976432]
  Plug-and-Play diffusion prior (DP) frameworks have emerged as a powerful paradigm for imaging reconstruction.<n>We present a novel approach to resolving bias-hallucination trade-off, achieving state-of-the-art gradients with significantly accelerated convergence.
  arXiv  Detail & Related papers  (2026-02-26T16:58:43Z)
• Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators [4.887201041798969]
  We introduce Proximal-IMH, a scheme that corrects samples from the approximate posterior through an auxiliary optimization problem.<n>For idealized settings, we prove that the proximal correction tightens the match between approximate and exact posteriors, thereby improving acceptance rates and mixing.<n>The method applies to both linear and nonlinear input-output operators and is particularly suitable for inverse problems where exact posterior sampling is too expensive.
  arXiv  Detail & Related papers  (2026-02-24T22:58:50Z)
• Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs [55.77845440440496]
  Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
  arXiv  Detail & Related papers  (2026-02-24T05:32:03Z)
• Supervised Guidance Training for Infinite-Dimensional Diffusion Models [47.65586147952857]
  In inverse problems, the aim is to sample from a posterior distribution over functions obtained by conditioning a prior.<n>We prove that the models can be conditioned using an infinite-dimensional extension of Doob's $h$-transform.<n>We propose a simulation-free score matching objective (called Supervised Guidance Training) enabling efficient and stable posterior sampling.
  arXiv  Detail & Related papers  (2026-01-28T16:39:39Z)
• Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations [57.179679246370114]
  We identify the distribution of random perturbations that minimizes the estimator's variance as the perturbation stepsize tends to zero.<n>Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length.
  arXiv  Detail & Related papers  (2025-10-22T19:06:39Z)
• Randomized Physics-Informed Machine Learning for Uncertainty Quantification in High-Dimensional Inverse Problems [49.1574468325115]
  We propose a physics-informed machine learning method for uncertainty quantification in high-dimensional inverse problems. We show analytically and through comparison with Hamiltonian Monte Carlo that the rPICKLE posterior converges to the true posterior given by the Bayes rule.
  arXiv  Detail & Related papers  (2023-12-11T07:33:16Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.