arXiv:2606. 04804v1 Announce Type: new 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.
By Jian Xu, Delu Zeng, John Paisley, Qibin Zhao
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.
The paper introduces D$^3$Opt, a derivative‑free diffusion‑based framework that separates constraint modeling from objective optimization in chance‑constrained programs. It learns a risk‑conditioned diffusion model once on constraint‑filtered decisions, then uses an annealed, particle‑based Feynman–Kac correction to optimize any post‑specified objective without retraining. The method is proven to preserve feasibility and provides an error bound, and experiments show effective optimization across smooth, non‑smooth, and non‑convex objectives, including objective transfer tasks.
By Ziwen Liu, Yan Liu, Congying Han, Tiande Guo, Yao Yan, Weichen Zhao
The paper introduces PDE‑Residual‑Informed Kriging (PRIK), a Gaussian process framework that embeds global PDE residual information directly into its covariance structure via a PDE‑residual kernel. PRIK yields a posterior mean with minimal integrated squared PDE residual and offers statistical interpretations linking variance to residual energy. The authors prove quasi‑optimality and convergence rates, and demonstrate the method’s PDE‑solving capability through numerical experiments.
By Yu Tan, Shichang Du, Yongxiang Li
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. 00703v1 Announce Type: cross Abstract: Low-precision pretraining (FP8, MXFP4, NVFP4) is now standard for frontier language models, yet the literature is almost entirely achievability -- algorithms and empirical scaling laws -- with no matching characterization of what is information-theoretically possible.
By Munsik Kim