Physics-Informed MeanFlow for PDE-Constrained Inverse Problems

Generative models have demonstrated promising results for inverse problems governed by Partial Differential Equations (PDEs). However, current diffusion-based methods can make limited progress at early sampling stages, when constraints are evaluated on the Tweedie posterior-mean estimate that regresses toward the dataset mean, and accurate reconstruction can require long sequential sampling trajectories. To address these, we introduce \textbf{P}hysics-\textbf{I}nformed \textbf{M}ean\textbf{F}low (PIMF), a framework that combines a staged PDE-residual fine-tuning with an iterative guidance algorithm that uses a MeanFlow projection as a clean-state plug-in, enabling more informative constraint evaluation throughout sampling. Doing so, we achieve three results: (i) improved accuracy and computational efficiency in solving inverse problems in most of the benchmark tasks, (ii) efficient ensemble-based uncertainty estimation, and (iii) the discovery of alternative physically valid solutions.