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
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: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
arXiv:2505. 22391v2 Announce Type: replace-cross Abstract: Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems.
By Yi Zhang, Peng Wang, Difan Zou
arXiv:2506. 20181v2 Announce Type: replace Abstract: We study operator relevance in data-driven partial differential equation (PDE) discovery.
By Ronald Katende