Condition-Aware Graph Flow Matching for Modeling the Distributions of Complex Fluid Systems

Accurately modeling the full distributions of possible states is crucial for understanding statistical properties and enabling reliable predictions in complex fluid systems. Recently, diffusion models and flow matching have shown promise in these tasks. However, they remain limited in uncovering the general principles of systems from multiple short trajectories across the condition space. In addition, they exhibit inferior adaptability to large irregular geometries, particularly in regions with sharp gradients. In this paper, we propose a condition-aware graph flow matching (CGFM) method that combines condition-aware flow matching with a hierarchical graph structure to learn the full distributions of fluid systems from incomplete training data. Specifically, CGFM constructs a flow enabling smooth interpolation across physical conditions and parameterizes the graph-conditioned vector field through HieraGraphNet. HieraGraphNet performs message passing across multilevel graphs to capture multi-scale dynamics and facilitate long-range information interactions in fluid systems. Moreover, we introduce a topology- and geometry-aware graph coarsening scheme that incorporates topological connectivity and local geometric density to construct reliable coarse graphs. We validate the effectiveness of CGFM on three canonical scenarios across both 2D and 3D dynamics, which demonstrate its superior performance compared with that of state-of-the-art baselines.

Many fluid dynamics problems require modeling distributions over possible flow states rather than making a single deterministic prediction. These distributions are essential for characterizing statistical properties and predicting system behavior under new physical conditions. This is challenging because high-fidelity simulations are expensive, training data may contain only short trajectory segments, and fluid domains are often represented by large irregular meshes. We propose a condition-aware graph flow matching (CGFM) method for learning equilibrium distributions of fluid states from incomplete simulation trajectories. CGFM represents fluid systems as graphs and learns how possible flow states change smoothly across different physical conditions. To better capture both local details and global interactions, our model uses a hierarchical graph network that passes information across multiple levels of resolution. We also design a geometry-aware graph simplification strategy so that the model can better handle irregular meshes and complex shapes. Experiments on 2D and 3D fluid systems show that CGFM improves distributional accuracy and generalization compared with existing baselines.