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
arXiv:2606. 02331v1 Announce Type: cross Abstract: Diffusion-based inverse problem solvers can produce realistic reconstructions, but realism alone does not ensure that the recovered details are supported by the measurement.
By Pengfei Jin, Yiqi Tian, Kailong Fan, Bingjie Qi, Quanzheng Li
arXiv:2608. 18004v1 Announce Type: new Abstract: Probabilistic modeling of physical fields benefits from both a data-driven prior and known physical structure such as the governing equations.
By Yixuan Sun, Anirban Samaddar, Sandeep Madireddy
arXiv:2606. 11650v1 Announce Type: new Abstract: Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation.
By Handi Zhang, Adrienne M. Propp, Brooks Kinch, Houman Owhadi, Nathaniel Trask
arXiv:2508. 09156v3 Announce Type: replace-cross Abstract: We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems.
By Jan Tauberschmidt, Sophie Fellenz, Sebastian J. Vollmer, Andrew B. Duncan
arXiv:2607. 21721v1 Announce Type: cross Abstract: Learned generative priors are increasingly used for ill-posed Bayesian inverse problems, their posterior uncertainty treated as earned from data.
By Ali Siahkoohi, Sina Alemohammad
arXiv:2605. 07565v2 Announce Type: replace-cross Abstract: We study Bayesian Optimisation (BO) in settings where the objective function is influenced by uncontrollable environmental contexts governed by an unknown probability distribution.
By Tigran Ramazyan, Denis Derkach