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.
arXiv:2603. 00393v2 Announce Type: replace-cross Abstract: Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data or inherent non-uniqueness.
By Ali Siahkoohi, Kamal Aghazade, Ali Gholami
FB‑GDM is a fully‑Bayesian guided diffusion method that eliminates the need for task‑specific hyperparameter tuning in linear inverse problems. It derives a closed‑form conditional score from a Gaussian approximation of ΦGDM, treating two precision parameters as latent variables inferred via variational inference at each reverse step. Experiments on CelebA‑HQ show that FB‑GDM outperforms ΦGDM at its nominal setting, matches a ground‑truth‑calibrated oracle within 0.1 dB, and remains robust to changes in the forward operator, noise level, or image distribution without hallucinations.
By Gatien S\'eguy (SATIE), Thomas Rodet (SATIE)
arXiv:2606. 02309v1 Announce Type: new Abstract: Generative models are increasingly used as priors for inverse problems, but their ability to produce realistic images creates a basic trust problem: a plausible reconstruction may be supported by the measurements, or it may be filled in by the prior along unobserved directions.
By Pengfei Jin, Na Li, Quanzheng Li
arXiv:2609.15702v1 Announce Type: new
Abstract: Pretrained score-based diffusion models provide strong unconditional priors, yet enforcing measurement or physics consistency in inverse problems is of...
By Zihao Wang
arXiv:2503. 14549v4 Announce Type: replace Abstract: Scientific generative models must turn tractable local decisions into globally correlated samples that respect physical constraints.
By Michael Chertkov, Hamidreza Behjoo, Sungsoo Ahn