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 make limited progress at early sampling stages: constraints are evaluated on the Tweedie posterior-mean estimate, which at high noise is close to the dataset mean and therefore carries little information about the specific solution. Moreover, accurate reconstruction can require thousands of guided denoising steps, each depending on the previous one. Both effects slow convergence. To address them, we introduce Physics-Informed MeanFlow (PIMF), which fine-tunes a MeanFlow prior with a PDE-residual loss and, at inference, evaluates observation and PDE constraints on the MeanFlow one-step estimate of the clean state rather than on the posterior mean, so that guidance is informative from the first step. PIMF (i) improves accuracy and computational efficiency over diffusion-based solvers on most benchmark tasks, (ii) makes ensemble-based uncertainty estimation affordable, and (iii) recovers alternative physically valid solutions consistent with the same sparse observations.