Physically-Informed Flow Matching with Graph Neural Networks for Complex Fluid Dynamics

Computational fluid dynamics (CFD) simulations traditionally require extensive computational resources, limiting their utility in many scientific and engineering applications at scale. We introduce Physically-Informed Flow Matching Graph Networks (PIFM-GN), a novel generative framework that directly samples fluid states under specified physical conditions without requiring expensive time-stepping simulations. The key innovation of our approach is the incorporation of incompressibility constraints directly into the flow matching transport process by parameterizing velocity fields through vector potentials, with graph-based curl operators ensuring divergence-free predictions without requiring global pressure-Poisson solves. Experiments on diverse fluid dynamics problems -- ranging from two-dimensional surface pressure distributions and complete flow fields, to complex three-dimensional airflow fields -- demonstrate that PIFM-GN generates high-fidelity samples with significantly fewer sampling steps than diffusion-based alternatives. Most notably, our model maintains competitive performance even with a single sampling step, a regime where diffusion models completely fail. Our generated samples accurately reproduce the statistical characteristics of target flows, successfully capturing multi-modal pressure distributions across various flow conditions, while achieving significant computational speedups compared to diffusion-based methods. PIFM-GN thus enables efficient generation of fluid states for downstream analysis and design tasks in scientific and engineering applications. The code is available at https://anonymous.4open.science/r/pifm-gn-F75F/.