POD-Flow: A POD-Driven Flow Matching Framework for Physical Field Generation on Irregular Geometries

Generative models, including diffusion and flow matching models, provide a promising probabilistic framework for solving forward and inverse problems governed by partial differential equations (PDEs), particularly when observations are sparse or noisy. However, most existing approaches rely on U-Net or Fourier-based architectures designed for regular grids, limiting their applicability to physical fields discretized on irregular meshes. We propose POD-Flow, a flow matching framework that incorporates a Proper Orthogonal Decomposition Neural Operator (PODNO) as its generative backbone. The data-derived POD modes form a globally supported basis that preserves the dominant long-range spatial correlations of irregularly discretized physical fields, enabling efficient global modeling without graph message passing or attention mechanisms. POD-Flow adopts direct final-solution prediction rather than conventional velocity prediction, a formulation that naturally aligns with POD projection: the dominant modes retain most of the physical-field energy while filtering out much of the broadly distributed noise energy, yielding high-signal-to-noise estimates early in inference.During inference, observation-guided sampling further steers the generated fields toward available measurements, allowing a single trained model to address forward prediction, field reconstruction, and parameter inversion under different partial-observation settings without retraining. Experiments on multiple PDE benchmarks demonstrate that POD-Flow achieves competitive or improved reconstruction accuracy while requiring substantially fewer sampling steps than diffusion-based baselines. These results establish POD-Flow as an efficient generative framework for probabilistic PDE solving beyond regular-grid discretizations.